Correct solutions were submitted by:
Bishop Hartley High School, Columbus, Ohio
Brian Hickey, Grade 12
Cheshire High School, Cheshire, Connecticut
Grade 10
Paul Laconte, Matt Zibell, Vanessa Fitch, Kevin Solli, Rosina
Pannone, Ted Petremont, Jennifer Rinslandm, Amanda Miller,
Mary Grelle, Don Kim, Caryl Anquillare, Jason Troske,
Lauren Cozzolino
Grade 9
Karl Ljungquist
Fairfield High School, Fairfield, Connecticut
Bilal Seyal, Kyle Halligan, Robert Eng, Kelly Nash, Grade 9
Granada High School, Livermore, California
Erica Campbell, Grade 10
Hanover High School, Hanover, New Hampshire
Seth Silverstein, Josh Fischel, Grade 12
Orin "Bang" Pacht, Grade 11
Hinckley/Finlayson High School, Hinckley, Minnesota
Angie Clark & Joanna Olson, Grade 10
Home Schooled, Akron, Ohio
Anna Margush, Grade 4
Homewood-Flossmoor HS, Flossmoor, Illinois
Stacey Stewart, Grade 10
Ignacio Senior High, Durango, Colorado
Deanna Owens & Kate Treanor, Grades 9 & 10
I.S. 119 Queens, New York
Jason Lee, Grade 8
Loreto College, Marryatville, South Australia
Therese Quinn, Beth Loveday, Year 10
Mount St. Joseph Academy, Flourtown, Pennsylvania
Grade 10
Lauren Grabowski, Karen Wing, Jenn Cody, Jill Sommer,
Lori Simmons, Tara Shepherd & Luisa Galdi
Grade 9
Katie Walder, Joanne Getson & Shannon Firth, Lauren
Goldbeck, Lindsay Parsons, Annie McIntyre & Liz Croney,
Celine McElwee & Amy Barbieri
Murray Junior High School, Ridgecrest, California
Cassie Gorish Grade 8
Thomas S. Kuo, Grade 7
Newport High School, Bellevue, Washington
Ryan Bagley, Dan Piha, Grade 9
Robert Crow, Travis Kilpatrick, Grade 8
Pleasant Hill HS, College Park, California
Chelsea Wogsland, Melissa Brown, Allen Song, Ben Tam, Grade 10
Ridgeview High School, Bakersfield, California
Resa Kim & Ryan Duncan, Grade 10
River Oaks Public School, York, Ontario, Canada
Hassan Shamji, Grade 6
Smoky Hill High School, Aurora, Colorado
Katie Crane, Jessica Whitehead, Grade 10
Tina Barber, Grade 9
Jennifer Sprangers & Yunny Chen, Grades 9 & 10
Sturgeon Bay High School, Sturgeon Bay, Wisconsin
Meagan Cihlar, Derek Mccarty, Grade 9
Toano Middle School, Berkeley, California
Marina Spitkovskaya, Grade 8
Walnut Hill School for the Arts, Natick, Mass.
Rachel Schneebaum & Ashima Scripp, Grade 10
Walter Williams High, Burlington, North Carolina
Dorothy Moorefield, Grade 11
Martin County High School, Stuart, Florida
Grade 12
Danielle P. Cegelis
Grades 10 & 9
Christine Francescani & Julie Conant, Dylan Matheson,
Tyson Howard & Andrew Irwin, Jon Eberst, Tim Casale &
Danielle Roberts, Julia Schumm & Edna Evans, Laura Ejups
Cathy Freihofer & Todd Spahn, Mike Schmidt & Phillip Pool
Rod Hofer, Christen Bogenrief & Melissa Sloane, Sara Holtzman
& Jenny Schaefer, Eric Petersen & Nick Tall, Jaime Uhazie,
Dustin Hicks & Kaitlin Coffinbarger, Celine McElwee &
Amy Barbieri
Celine McElwee & Amy Barbieri
Grade 9
Mount St. Joseph's Academy
To do this problem we decided that we were first going to draw
ourselves a few diagrams to help us figure this out. First we drew a
diagram of you and your shadow and filled in the measurements we knew to
solve for what we didn't know. We then figured that the tangent of the
angle of elevation of the sun(x) was equal to 70" divided by 90". Using
reverse tangent we were able to find that the angle of elevation of the sun
was approximately 37 degrees. Knowing this we plugged it into our diagram
of the tree and its shadow, so we now knew that the shadow is 920" and the
angle of elevation of the sun was approximately 37 degrees by using
tangent. Therefore, the tree is approximately 685 inches or 57'1". If you
were 6' tall, the tree would still be short enough because it would be
approximately 705" or 58'9".
*****************************************************
Chelsea Wogsland
I had to find the relation between the height of the tree to the length
of its shadow in order to answer the question. But to do this, you must
use the height to distance relation of your shadow. First, I converted all
measurements into inches. Then I divided the shadow length - 94" by your
true height - 70".
This yielded 1.343. Then I divided the tree's shadow- 920" by 1.343,
which equaled 585.136. Divided by 12(12"), it came out to about 57'1".
Therefore it shouldn't hit the lilac. I used the same basic process to
account for the work boots. The tree would then still fit, at about 58'7",
but by a much tighter margin.
*****************************************************
Melissa Brown
If she is five feet ten inches and her shadow is 94 inches the first step
you should take is to convert her height to inches. This gives her the
ratio 70" =94" which breaks down to 1"-1.342857" by dividing each side by 70.
The next step is to convert the trees shadow from 76 feet 8 inches to 920
inches. To find the trees height by using the number 1.342857 from the first
ratio you will get 685.106 inches. Then by dividing the nunber by 12 you
will get the height is less the 59 it will not hit the lilac bush.
If her height were closer to 6 feet ( 72 inches) the ratio for 72'=94
would be 1"=1.3055". Then 920" divided by 1.3055" = 704.71". Divided by
twelve it is 58.7 feet which is also less than 59 feet.
*****************************************************
Allen Song
In solving this problem, I first listed the given data. I converted the
feet into inches because I thought it would be easier to work with whole
numbers. With these numbers, I found that I could get a ratio by dividing
the woman's shadow length by her actual height. 94 divided by 70 = 1.34285.
Next, I divided the length of the tree's shadow by the ratio, I found.
This gave me the trees height in inches. 920 divided by 1.34285=685.106.
Then I divided the height ( in inches) by 12 in order to get the height
in feet. 685.106 divided by 12= 57.092.
The results told me that the tree was too short to hit the bush.
If she was 6 feet in her working boots, and her shadow was still 94
inches, I would divide 94 by 72.
With this ratio, I could also find out the new height of the tree. I
would simply divided the tree's height by the new ratio.
*****************************************************
Ben Tam
The first thing I did was I converted the information given from feet
into inches. I started with the measurment of the trees shadow. I
multiplied 76 feet by 12 and got 912. I then added the eight inches
and the shadow length of the tree was 920 inches. I then converted the
five feet ten inches measurement into inches by taking 5 and multiply
it by 12 and then adding 10 to it. I got 70 inches. I took the shadow
measurement of 94 inches and dividing it by 70. I got 1.34285. This
means for every one inch, the shadow length would be 1.3428 etc. The
formula for finding out the heght of the tree is shadow length divided
by 1.342 etc. I did this 920 divided by 1.34285=685.106. I took that
number and divided it by 12 and got 57.092 feet. Since it is less than
59 feet, it wouldn't hit the lilac bush.
Now, for the next problem, I have to consider the fact that you are
wearing boots which makes her 2 inches taller. I multiplied 6 feet by
12 and got 72 inches. I divided that number by 94 and got 1.305.
For every inch, the length of the shadow would be 1.305. I used the
formula of shadow length divided by 1.305 = height of tree. I got 704.68
and divided that by 12 to convert it into feet. I got 58.723. Since
it is less than 59 feet it would also miss the lilac bush.
This means you can cut it down, but the tree would fall awfully close
to the lilac bush.
***********************************************
From: Marina Spitkovskaya
Grade: 8
School: Toano Middle School
Answer: In either situation, the tree will not fall on the lilac bush.
***********************************************
Christine Francescani & Julie Conant
Martin County High School
Grade 10
Moroff88h@aol.com
To begin, we drew the diagram and labeled points on the two triangles
A-E. We said that triangle BAC is similar to triangle EAD because
corresponding angles are congruent, and the corresponding sides are
in proportion. We then said that side CA is to side CB as side DA is
to side DE. Therefore, 94" is to 70' as 920" is to X. So, X equals
685.11", which is 57.11'. If the girl is 6' tall, then the tree is 58.7' tall.
In either case, the tree will not hit the lilac bush.
***********************************************
From: dylan matheson
Grade: 10
School: martin county high school
Answer: Hi - I love this place. this problem was perfect for my geometry
class this week. we were learning about similar triangles so I simply
plugged all the given numbers into a drawing and it
was easy. Here is the drawing. I hope it turns out ok.
A
\
\ \
\t \
\r \ B
\e \
\e \y \
\ \o 70" \
\______________________________\u______________________ \C
E D
\------------826"--------------\-------------94"---------\
\-----------------------------920"------------------------\
These are the given numbers in inches for the first question where you are
5'10". Using the proportion- 94/920=70/x and solving for x which is
the height of the tree you get that the height of the tree is 57'1" and it won't
crush your lilac bush. Now if you were 6' or 72 inches then
according to my equation
there would be 58'8" and it would not crush your precious lilac bush.
Thank you - hope this all turns out - good bye...
***********************************************
Ryan Bagley, Newport High School
First I found the ratio of your height, 5'10", to your inches given, then I
found out how many inches were in the tree and cross multiplied with
those numbers and came up with the tree being 57 feet. Then I did the
same for you as you being 6'0" and came up with the tree being 58.7
feet. The tree will feet if you are 5'10" or 6'0", it doesn't matter.
***********************************************
From: Dorothy Moorefield
Grade: 11
School: Walter Williams High
from_email: dawalrus@netpath.net
Answer: To solve this problem, use the fact that the angles formed from
revolving from the ground to the top of Annie and the tree (x)
are congruent. Too similar right triangles are formed.
tree Annie
/l /l
(y^2+920^2)^0.5" / l 13736^0.5" / l 70"
/ l y" /x l
/x l -------
--------- 94"
920"
The lengths of the bases of the triangles formed are the lengths
of the shadows converted to inches. The height of the tree is
unknown so it is called y. Annie's height is 70 inches.
The lengths of the hypotenuses are found with the Pythagorean
theorem [leg1^2+leg2^2=hypoteneuse^2]
Set these triangles in circles whose radii are the hypotenuses
to find the length of y. Since x=x the cos x = cos x.
The cos x = (the length of the adjacent side)/(the length of the
hypotenuse)
cos x = 920/((y^2+920^2)^0.5) cos x = 94/(13736^0.5)
920/((y^2+920^2)^0.5 = 94/(13736^0.5)
11626150400 = 8836y^2 + 7478790400
4147360000 = 8836y^2
4147360000/8836 = y^2
y is approximately = 685.106" approximately = 57'
So the tree could come down within 2 feet of hitting the lilac
bush.
If Annie is closer to 6' plug in 72" for her height and repeat
the process. DO NOT FORGET TO CHANGE THE LENGTH OF THE HYPOTENUSE.
cos x = 920/((y^2+920^2)^0.5) cos x = 94/(14020^0.5)
920/((y^2+920^2)^0.5) = 94/(14020^0.5)
11866528000 = 8836y^2 + 7478790400
4387737600 = 8836y^2
4387737600/8836 = y^2
y is approximately = 704.681" approximately = 58.724'
If Annie is closer to 6', the tree could come down but it would
be cutting it awful close.
An easier way to solve this problem would have been to set up
proportions because the triangles are similar. However, the cos x
method was more fun to explain.
***********************************************
From: Erica Campbell
Grade: 10
School: Granada
Answer: Yes, you will be able to cut down the tree without it hitting the lilac
bush. The tree is 57.09 ft tall if you measured your shadow without your boots
on and 58.7 ft tall if you measured your shadow with your boots on. But either
way the tree is less than 59 ft tall so you will be able to cut it down without
it hitting the lilac bush
***********************************************
Tyson Howard and Andrew Irwin, grade 10
Martin County High School
Stuart, FL
The tree will be 57 feet or 58.7 feet tall. Those are
both under 59 feet, so the lilac bush won't be smashed. A
proportion problem was used to get these figures. First, I
converted all feet to inches. Then I set the tree's shadow
over x, and that would equal your shadow over your height.
The actual equation used was 920 inches/x = 94 inches/70
inches. If your 5'10", I got 685 inches or 57 feet. If
your 6'0", I got 705 inches or 58.7 feet.
***********************************************
Jon Eberst, 10
Martin County High School
Stuart, Florida
I have concluded that the tree will not land on the lilac tree either
time because you are 57 feet with out boots on and you are 58 feet with
boots on.
From, Jonathon T. Eberst Sr.
P.S. Can you please do some problems about fishing? I, and many other
students at my school would appreciate it.
***********************************************
Rod Hofer, grade 9
Martin County High School
Stuart, Florida
In order to solve the problem, the first thing I did
was change all the numbers into inches. Once I did that, I
set up this ratio: lady tree
shadow 94" 920"
----- -----
real height 72" ? (=685",57')(without boots)
To find out my unknown number, cross multiplied and
then divided. I got 685".
With boots it would be 58'8". It is close to 59' but
you should be o.k.
***********************************************
Danielle P. Cegelis, grade 12
Martin County High School
Stuart, Florida
ANSWERS:
1)When considering your height without the workboots the tree will not come
out as being taller than 59 feet.
2)When considering your height with the boots the tree is still short
enough that it will not hit the lilac.
EXPLANATION:
1)I drew a picture of the tree and its shadow and compared it to a picture
drawn of you as compared to your shadow. These pictures are of triangles
which are similar therefore they can be used in proportion with one
another. I converted the measurements to inches for easier calculations. My
conversions were: a) your height: 70"
b) your shadow: 94"
c) the trees shadow: 920"
***********************************************
CHRISTEN BOGENRIEF, grade 9
MELISSA SLOANE, grade 9
MARTIN COUNTY HIGH SCHOOL
STUART FLORIDA
TO FIGURE OUT HOW TALL THE TREE IS YOU MAKE TO SIMILAR
TRIANGLES. YOU THEN MAKE A RATIO MINE WAS
920 INCHES (THE TREES SHADOW):94 INCHES (YOUR SHADOW) I THEN
MADE THE RATIO X(THE TREES HEIGHT):70 INCHES(YOUR HEIGHT,
5FT 10 INCHES) I THEN TOOK THESE TWO RATIOS AND SET THEM
EQUAL. NEXT I CROSSED MULTIPLIED 920 TIMES 70 AND 94 TIMES X
YOU THEN GET THAT 94X IS EQUAL 64400. SO X(THE TREES HEIGHT)
EQUALS 685.1 OR 57 FT 1 INCH. I DID THIS SAME THING WITH YOUR
HEIGHT BEING 6 FT IN YOUR BOOTS AND GOT THAT THE TREE WOULD
ONLY BE 58 FT 8 INCHES. SO GO AHEAD AND CUT DOWN YOUR TREE IT
WILL NOT HURT YOUR LILAC BUSH. IT SHOULD MISS IT BY 4
INCHES.
***********************************************
Cathy Freihofer, grade 9
Todd Spahn, grade 10
Martin County High School
Stuart, Florida
The answer to your problem about the tree is that either way if you
fell the tree it would not hit the bush.
Because the triangle created by your height, your shadow, and the
tip of the shadow to the top of your head is similar to the height of the
tree, the shadow of the tree, and the top of the tree to the tip of the
shadow. We can put those numbers into proportions equal to each other.
94/920=70/x
When we cross multiple and then divide we can get the solution to x
or the height of the tree.
Both times the tree is shorter then the distance between the tree
and the bush.
***********************************************
Mike Schmidt, grade 9
Phillip Pool, grade 10
Martin County High School
Stuart, Florida
The answer to the question about the length of the tree is that
yes, at the height it is now, it will not hit the lilac bush if you fell
it. Also if your height was more than what you said the first time, then it
will also be fine if you fell the tree.
The method that we used to get the answer to this question is
simple. We took the length of your shadow, in inches, and put it over the
length of the tree's shadow like this- 94/920. We then put an equal sign
and put your height over "x" in another fraction- 70/x. We let "x" be the
height of the tree.
Since the triangle created by your shadow, your height, and the tip
of the shadow to the top of your head, is similar to the triangle created
by the shadow of the tree, the trees height and the top of the tree to the
tip of the shadow, the proportions would be equal. because of that we can
cross-multiply and then divide to get the solution to "x".
In both cases the tree is shorter than 59 feet, so it would not hit
the bush if you cut it down.
***********************************************
Sara Holtzman-9th
Jenny Schaefer-9th
Martin County High School
Stuart Florida
First we must convert feet into inches. Six feet
converts into 72 inches. 76'8" converts into 920 inches.
Then we set up a proportion 920":94"=x:72" reducing it to
460":47"=x:72". Multiplying it out we came to the equation
33,120"=47"x. Dividing the equation we came to the
conclusion that x=704.68085". Converting this back to feet
we got 58.723404 feet which rounds off to 59 feet. This
tells us that the tree would just miss the lilac bush. We
didn't figure it out without your shoes on because most
people wouldn't go outside with no shoes in the dead of
winter.
/tree
/ |/
/ |
/ |
/ |
you\ / |
/| |
/ |6" |
/ | |
-------------------
|------|
94"
|------------------|
76'8"
***********************************************
Robert Crow
Newport High School
Mr. Mabbott
Grade 8
Both work. Without your workboots, the tree is 57'1", and with
your workboots, the tree is 58'9". When you drop the tree, it won't fall
on the lilac bush.
I got my answer by first changing all the height in feet into
inches, and then setting up ratios that looked like:
n 70 n 72
--- = --- without boots and --- = --- with boots
920 94 920 94
The first one ended up being n=685 in., which is 57'1", and the
second one ended up being n=704.68 in., which is 58'9", so the tree won't
fall on the lilac.
***********************************************
From: Brian Hickey
Grade: 12
School: Bishop Hartley High School
from_email: bhickey@cd.pvt.k12.oh.us
Answer: With or without your boots on, the tree will not land on your
lilac bush (unless an act of God intervenes).
I solved the problem through the use of proportions.
I set your height over your shadow's size and set that equal to
the tree's height (t) over its shadow. And since the bush is
59' away, then (t) must be less than 708".
59'=708"
5'10"=70"
70/94 = t/920
6'=72" 94t=64400
76'8"=920" t=685.106383
The tree falls short if weren't wearing your boots.
72/94 = t/920
94t=66240
t=704.6808511
Again the tree falls short even if you wore boots.
Either way you'll get to enjoy your lilac bush.
***********************************************
From: Thomas S. Kuo
School: Murray Junior High School, Ridgecrest, California
Grade: 7th
POW February 5 - 9, 1996
I set up a proportion to solve these two problems.
(1) Let x = height of tree
Convert units to inches: 76'8" = 920", 5'10" = 70"
Set equation as x/920 = 70/94, then x = 685.11" ~ 57.09'
The tree is about 57.09' tall which is less than the maximum distance
allowed, 59'. It won't land on the lilac.
(2) Let x = height of tree
Convert units to inches: 6' = 72"
Set equation as x/920 = 72/94, then x = 704.68" ~ 58.72'
The tree is about 58.72' which is less than the distance of 59'.
Your lilac is safe.
***********************************************
Angie Clark & Joanna Olson grade 10 Hinckley/Finlayson High School
You will be able to cut down the tree without it falling on the
lilac bush. The tree is about 57 feet tall. We got this answer by
setting up similar triangles (see attachment JO & ANG). Our
equation looked like 70/x=94/920. Our x equals 685.10638". That
converts to about 57 feet. When we raised your height to 72" (your
boots) it changed the answer to 704.68085". That converts to almost
59 feet. You probably won't cut the tree even with the ground, so
it still won't hit the lilac bush.
***********************************************
From: Katie Crane
Grade: 10
School: Smoky Hill
Answer: With or without the boots the tree will just fit. It's either
57'1" (without the boots) or 58'9" (with the boots)
***********************************************
From: Tina Barber
Grade: 9
School: Smoky Hill High School
Answer: 94/920=70/x --Proportion between Ann's shadow (94 in)
and the tree's shadow (920 in) and Ann's height
(70 in) and the tree's Height (x)
cross multiply---64,400=94x
64,400/94=x
685.11 in =x 57.09 ft =x
if Ann is 6 ft (72 in) set up new proportion 94/920=72/x
cross multiply---66,240=94x
66,240/94=x
704.68 in =x 58.7 ft =x
**Whether Ann is 5'10" or 6' the tree is still under 59".
Chop Away!
***********************************************
From: Orin "Bang" Pacht
Grade: 11 (junior)
School: Hanover High School
Answer: If your boots make you six feet tall, then you will miss the lilac
bush by about 3.3 inches, therefore, the tree can successfully be
cut down without wrecking anything (not to mention the fact that
you probably wouldn't be cutting the tree at the ground).
The 3.3 inches I gave to play with was the difference between
the calculated height and the distance from the tree to the lilac bush. I
didn't do anything with your 5'10" height because I figured that it didn't matter
since I used the larger height, therefore making the tree come out as tall as
possible.
To find the
length of the tree after it has been felled, you have to compare the length of
the two shadows and your height. In order to do this, you have to set up the
three knowns and the unknown in two ratios. This may be the lengths of the
shadows on top and the actual heights on the bottom or the measurements
related to you on top and the measurements related to the tree on the bottom.
The order of the math is to multiply the number that is above or below the
unknown (depending on how the problem is set up) by the number in the other
ratio on the same side as the unknown--a.k.a. cross-multiplying. Divide that
product by the remaining known and that quotient is the answer.
gotta go
Bang
***********************************************
From: Jennifer Sprangers,& Yunny Chen
Grade: 9,10
School: Smoky Hill High School
Answer: Solution: Since you need to drop the tree between the veggie garden
and the willow tree and without smashing the lilac bush(that is 59'
away) I figured out the height of the tree from it's shadow. If you
are 5'10" and your shadow is 94", then the ratio of your height to
your shadow is 70"/94". If the trees shadow is 76'8"(920"), then you
could form the proportion that 70"/94"= x/920".(x= the trees height).
Then you would cross multiply and divide to get the height of the tree =
57'1". This would be a good 2' away from the bush and would be safe to cut
down. But if you are close the 6' in your hiking boots then the
proportion would change to 72"/94"= x/920". If you cross multiply,
then divide the tree's height would be 58'8.67" Now I don't think
that it is so safe to cut down the tree with only 3" away from the
lilac bush. It would be too close for comfort, and considering the
way a tree falls, which isn't straight down, I think that the tree
would hit the lilac bush.
***********************************************
From: Seth Silverstein
Grade: 12
School: Hanover High School
Answer: Do to this problem I set up a proportion and did everything in inches.
The mans shadow was 94 inches and his height was 72 inches. The tree's shadow
was 920 inches and the problem was to find the height of the tree.
After solving the proportion, I found that the tree is 705 inches. Then,
converted to feet, it ends up being almost 59 feet tall. Therefore, he can
cut down the tree.
If you are standing 5'10" tall, the tree ends up being about 57 feet tall.
Seth Silverstein
***********************************************
From: Cassie Gorish
Grade: 8
School: Murray Junior High
Answer: If you are 5'10", then:
_920"(tree shadow)_ = _94"(shadow)_
X (tree height) 70"(your height)
Cross multiply.
32200" = 47x
685.10" = x
57' = x
The tree is about 57 feet high. It won't hit the lilac bush.
If you are 6' with boots, then:
_920"_ = _94"_
X 72
Cross multiply.
33120" = 47x
704.68" = x
57'7" = x
The tree is about 58'7".
***********************************************
From: Jason Lee
Grade: 8
School: I.S. 119 Queens
Answer: Okay, now:
Without the workboots:
76'8" = 920" 5'10" = 70"
Ratio of your shadow to tree's shadow = Ratio of your height to tree's height
94" : 920" = 70" : x" 64400 = 94x
| | 64400 | | ----- ---
| ---------- | 94 94
-------------------
94x 32200
----- = x
47
32200/47 = 685 5/47" = 57' 13/141"
The tree will not land on the lilac bush if you were 5'10"
Now, with the workboots:
76'8" = 920" 6" = 72"
Ratio of your shadow to tree's shadow = Ratio of your height to tree's height
94" : 920" = 72" : x" 66240 = 94x
| | 66240 | | ----- ---
| ---------- | 94 94
-------------------
94x 33120
----- = x
47
33120/47 = 704 32/47" = 58' 34/47"
The tree will not smash the lilac bush but it would come close to it.
-Jason Lee.
***********************************************
From: Resa Kim & Ryan Duncan
Grade: 10th
School: Ridgeview High School
Answer: It would work when she was 5'10" because the tree would be 57 feet tall.
It would also work if she was 6 feet tall because the tree would only be 58.7
feet tall.
***********************************************
Jessica Whitehead
Grade: 10
School: Smoky Hill High School
When you are 6' 0":
I tried the problem using tangent instead of sine.
tangent x = 72/ 94
x = approx. 37.45
I substituted 37.45 for x in the equation: tangent x = y/ 920,
so I have tan 37.45 = y/ 920. y = 704.68; then, divide by 12 to get
58.7 feet. Multiply 0.7 by 12 to get 8 inches.
So, if you are 6' the height of the tree is 58' 8"
When you are 5' 10":
tangent x = 70/ 94
x = approx. 36.67
I substituted 36.67 into the equation: tan x = y/920.
Tangent 36.67 = y/ 920
y = 685.10, divide by 12 to get approx. 57'.
When you are 6', the height of the tree is 58' 8"; when you are 5' 10",
the height of the tree is 57'. Either way, you will not hit the lilac bush.
***********************************************
From: Josh Fischel
Grade: 12
School: Hanover High School
Answer: When you are standing 5'10", the tree is 685 inches tall, or roughly 57
feet. This is less than 59 feet. No need to worry.
When you are standing at six feet tall, the tree is 704.7 inches tall, or 58.7
feet tall. That means that the tree still won't hit the lilac bush, but it'll
be less than four inches away from it.
Happy gardening!
Josh
***********************************************
Therese Quinn
Year 10
Loreto College Marryatville
South Australia
! \
! \
! \
! \
! \
! \
! \
! Height \
! of \
!Person \
! 70" \
! \
! \
! \
!____________________________ \
Length of shadow
(a)
To figure out the height of the tree, I first worked out the sun's angle of
elevation, by looking at the height of the person and the length of their
shadow, and using the trigonometry function 'tangent'.
tan a = opp
---
adj
= 70
--
94
tan 36.67 = 70
--
94
The sun's angle of elevation is 36.67 degrees.
Because the shadow of the tree was taken at the same time as the person's,
the sun's angle of elevation would have been the same. I used the tangent
function once again to figure out the height of the tree. (76'8" = 920")
tan 36.67 = height of tree
--------------
920
height of tree = tan 36.67 X 920
= 685"
= 57'1"
Therefore the tree wouldn't squash the lilac bush when it fell.
IN WORKBOOTS, THE PERSON IS LIKELY TO HAVE BEEN CLOSER TO 6'.
If this were the case, the numbers would have been different.
tan a = opp
---
adj
= 72
--
94
tan 37.45 = 72
--
94
Therefore, the sun's angle of elevation is 37.45 degrees
tan 37.45 = height of tree
--------------
920
height of tree = tan 37.45 X 920
= 705"
= 58'9"
In this case, the tree would only miss the lilac bush by 3". Personally, I
wouldn't risk it, but then, if the tree's really that ugly, and if it's shading
the garden.
***********************************************
Name: Beth Loveday
Year: 10
School: Loreto College Marryatville
State: South Australia
! \ ! \
! \ ! \
! \ ! \
! \ ! \
! \ ! \
! \ ! \
! \ ! \
! \ ! \
Height ! \ Height ! \
of person ! \ of tree ! \
70" ! a\ "x" ! \
!________________________\ Angle of !____________\
Length of Elevation Length of
Shadow 94" Shadow 920"
To figure out the height of the tree, I first figured out the sun's angle of
elevation. I did this by using the person's height and the length of their
shadow and then using the trigonometry function "tangent".
Tan= opp.
adj.
Tan a= 70/94
Therefore the sun's angle of elevation = 36.67 degrees.
Because the measurements were taken at the same time of day of the day, the
sun's angle of elevation was the same for both figures. So to find the height
of the tree I again used the "tangent" function.
76'8"= 920"
Tan= opp.
adj.
Tan 36.67= x
920
x= 36.67tan*920
x=684.99
Therefore the height of the tree is 57.08'
In this case you would have approximately 23" of space between the tree and the
lilac bush.
Considering the fact that the gardener was wearing work boots, her height would
be closer to 6 feet. In this case the numbers will be altered:
Tan= opp.
adj.
Tan a= 72
94
Therefore the sun's angle of elevation is 37.45 degrees
Tan= opp.
adj.
Tan 37.45= x
920
x= 37.45tan*920
x=704.66"
Therefore the height of the tree would be 58.7'
This meant that the tree would only miss the lilac bush by 3 inches. Which I
think is a bit risky!
***********************************************
Dan Piha Grade 9, Newport High School
After my research, I found out by your measurements that if you
drop the tree, it will not land on your lilac bush. I figured this out
by first I took your 5'10" and the 76'8" of the trees shadow and turned
them into inches. Than I made an equation with 70" over 94" equaling x"
over 920". So I times 94 by x and 70 by 920 and got 94x=64400. I
divided 94 and got 685". I found out that was equal to 57'1" which is
less than 59'. Next it said that you were 6' tall in your boots which is
72". I used the same problem as before, except I used 72" instead of
70". I soon found out that the total was 705" which is equal to 58'9".
That is less than 59'. So now you know that you can drop the tree.
***********************************************
H equals the height of the tree.
H over 70 inches = 920 inches over 94 inches.
cross multiply.
920 X 70 = 64,400. Divide 64,400 by 94.
H = 57 feet, 1 inch.
(In workboots.)
H over 72 inches = 920 inches over 94 inches.
cross multiply.
920 x 72 = 66,240. Divide 66,240 by 94.
H = 58 feet, 8.7 inches.
Stacey Stewart
Sophmore
Homewood-Flossmoor High School
***********************************************
From: Rachel Schneebaum & Ashima Scripp
Grade: 10
School: Walnut Hill School for the Arts
Answer: If the person is 5'10" the tree would not touch the lilac bush because
it is 57 feet approx.
(Height of person)70:94(shadow)=(tree)x:94(tree's shadow)
Therefore x = 685 in = 57 ft (approx.)
If the person is 6 ft the tree would fit but only by 3.3 in because
the height of the tree is 58.7 ft approx. because,
72:94 = x:920
Therefore x = 58.7ft approx.
***********************************************
From: Deanna Owens Kate Treanor
Grade: 9, 10
School: Ignacio Senior High
Answer: We figured out The problem by creating similar triangles. After we got
the similar triangles we were able to come up with an equation. We first
converted the feet into inches.
5'10" Solution,(X= the height of the tree): X/920=70/94 (cross multiply)
94X=64400 (divide both sides by 94) X~685.1 (convert inches back into feet)
X~57.1
6' Solution, (covert into inches again)(X=the height of the tree): X/920=72/94
(cross multiply) 94X=66240 (divide both sides by 94) X~704.7 (convert inches
back into feet) X~58.7
Therefore it did not hit the lilac bush for either situation.
***********************************************
Lauren Grabowski
Mount St. Joseph Academy
POW 2/5-9/96
When we first read this problem, we thought we would have to use
tangent, cosine, or sine; however, there turned out to be a much simpler
method. First, we converted all the measurements to inches by multiplying
the amount of feet by 12, since there is 12 inches in a foot, and then
adding the remaining inches. So the tree's shadow was 920 in. , your shadow
(as already indicated in the problem) was 94 in., your height without
boots was 70 inches, and with boots was 72 inches. We let the tree's height
be "x" . We also converted the distance between the tree and the lilac
bush and got 708 inches, so the height of the tree must be less than 708
inches. Next, to find the tree's height when you were without boots, we
created a proportion. Of course, to create a proportion, the two triangles
formed (one triangle is with the tree, shadow, and distance between the
tree top and end of the shadow as sides, and the other as you, your shadow,
and the distance between the top of you and your shadow's edge) are
similar. They are similar by AA Similarity Postulate. Assuming you and the
tree stood perpendicular to the ground, both have triangles right angles.
And assuming you measured both the shadows at the same time, both
triangles share similar angles from the sun's angle of depression.
your height 70": your shadow 94":: x the tree's height : 920" the tree's
shadow
70 * 920 = 6440/94 makes x= about 685 inches. It doesn't matter
that we rounded, just as long as the tree's height is less than 708 inches.
So if you were not wearing boots the tree will not crush your lilac bush.
To see if the tree would land on your lilac bush if you were wearing
boots which made you taller. We again set up a proportion.
your height with boots 72" :your shadow 94" ::x the tree's height:
920" the tree's shadow
72*920=66240/94 makes x= about 705 inches. Again, this measurement is
less than 708 inches (barely), so whether you were wearing your boots or
not when you measured the shadows, you can cut the tree down without
crushing the lilac bush.
***********************************************
Karen Wing
Grade 10
Mt. Saint Joseph Academy
P.O.W. 2/5-9
Because you are 5'10" and your shadow is 94", by taking the inverse tangent
of 70" / 94", you can find that the angle of the shadow is 36.7 degrees.
Since you were standing next to the tree the triangle between you, the end
of your shadow, and the top of your head is proportional to the triangle of
it, the end of its shadow and the top of the tree. Therefore the angle at
the end of the tree's shadow is also 36.7 degrees. If you then take the
tangent of 36.7 degrees and multiply it by the 76' 8" or 920", the length
of the shadow. The answer given to us is the length of the tree which is
57' 2". With this answer the tree will not hit the lilac bush but actually
be 1' 10" short of hitting it.
Since you are 6' in your boots or 72" and your shadow is 94", by taking the
inverse tangent of 72"/94" we find out that the angle is 37.5 degrees. By
using the same two proportional triangles you can say that the angle at the
end of the tree's shadow is now 37.5 degrees. By taking the tangent of
37.5 degrees and multiplying it by 920" again, we find that the length of
the tree is now 57' 9". This length of the tree will also be short of
hitting the lilac bush by 1'3".
***********************************************
Katie Walder
Problem of the week
Mount St. Joseph Academy
February 5-9, 1996
After looking at this problem, I decided to set up a proportion. I set
your shadow over the tree's shadow, which was 94/920, and I set that equal
to your height over the tree's height, which was 70/h. (I used your height
in your workboots first because if it fell without hitting the lilac bush
then, it must also fall without hitting it when you are at your normal
height.)
94/920 = 70/h
920 * 70 = 94h
64400 = 94h
h is approximately 685 in. or 57.09 ft.
Since it falls right when you have your workboots on, it must also fall
right when you don't.
***********************************************
Jenn Cody
Mt. St. Joseph Academy
Grade 10
POW 2-9
The tree will be under 59 feet using both of your heights, s o the tree
will not hit the lilac bush. I figured this out by using right triangle
trigonometry.
I figured out the angle of elevation of the sun by using your height and
the length of your shadow in the tangent ratio. I then applied that angle
of elevation to figure out the tree's height by using the angle of
elevation and the length of the tree's shadow in the tangent ratio.
In doing this for both of your heights I received the following:
Angle of elevation is about 37.5 degrees when you are 6'0" tall and about
36.7 degrees when you are 5'10" feet tall.
The tree's height is about 58.8 feet tall when you are 6'0" tall and about
57.1 feet when you are 5'10".
By find these heights, I found that the tree will just barely miss the
lilac bush.
***********************************************
Jill Sommer
Mt. St. Joseph Academy
Grade 10
POW 2/9
To get the answer for this problem of the week, I set up a proportion:
your height your shadow
----------- = -----------
(x) height of tree's shadow .
the tree
This proportion worked because the right triangles formed by the angle of
elevation of the sun, the shadows, and the heights of the tree and you were
similar.
If I used your height 5'10" or 70", the tree was only about 57'. You would
be okay to put the tree where you wanted to if it wasn't going to grow
much.
When I used your height of 6' or 72", however, the tree was almost 59'
(58.75' to be exact). This was cutting it a little close. I would
reconsider putting the tree in if it was this tall and you wanted to save
those lilacs.
***********************************************
Joanne Getson and Shannon Firth
Mount St. Joseph Academy
Grade 9
Problem of Week 2/5 to 2/9
The tree does not land on the lilac. For this problem we realized
that we had to find the height of the tree. First we converted all the
figures into inches, by multiplying anything that was in feet by twelve
(and then adding on the extra inches already given to us, if necessary). We
realized that two similar triangles are formed. The legs of the first
triangle are made up of the tree and it's shadow. The legs of the second
triangle are made up of you and your shadow. Since you know the length of
your shadow, and your height, and the shadow length of the tree, you must
find the height of the tree, by first finding the angle that is opposite
your shadow. Luckily we just finished a chapter on tangent ratios. So this
was easy. We used the ratio that the tangent of the angle of the sun = the
opposite side(the length of shadow) over adjacent(height of the object).
It looks easier like this:
tangent of angle of sun = length of shadow
height of the object
By using the tangent ratio of opposite side over adjacent side we could
find the angle of the sun. Our ration was tangent of x angle = 94 in over
70 in. The angle of the sun is 53.3 degrees. Now that we knew the angle of the
sun, we could use that angle to find the height of the tree. Our ratio was
tangent of 53.3 degrees = 920 in over x. X here was the height of the tree. In
this case x turned out to be 685in or 57.1 ft.
Then you brought up the fact that you were 6'. We took the same
steps as before. Now the angle of the sun was 52.5 degrees. Then we found
the height of the tree which was 705.94 in or 58.57 ft. In this case the tree
is close to the limit of 59 ft so you'd better cut the tree now before it
grows!
***********************************************
Lori Simmons
Problem of the Week
February 5-9
LSPOW2/9
Mount Saint Joseph Academy
Grade 10
When she is 6' --- To find the height of the tree, I first noticed
that the trees are similar by AA Similarity Postulate. Then I set up the
proportion 94:920::6:x. Then I cross multiplied and found the height of
the tree to be 58.83' high. Then, knowing that the tree was 58.83', I
subtracted that from 59 and knew that the tree was just short of falling on
the lilac.
When she is 5'10" --- I set up the proportion 94:920:: 5.83:x. At
this height, the tree would be 57.059' tall and it would be 1.94 short of
falling on the lilac.
***********************************************
Lauren Goldbeck
9th Grade
Mount St. Joseph Academy
February 5-9 Problem of the Week
I began solving the problem of the week by setting up a proportion of the
tree's shadow to your shadow ( 920/94) and then x (the tree's height) to
your height (x/70). I solved for x and I found that x is about 57.1 so if
you are 5 feet 10 inches, the tree will not destroy your lilac bush. If
you are 6 feet in height because of your boots the tree still won't hit.
I found this by setting up the same proportion of the tree's shadow over
your shadow and then x ( the tree's height) over your new height (72
inches). The proportion was 920/94=x/72. I solved for x and I found that
the tree's height is about 58.7 feet and this is under 59 feet so your
"ugly" tree will not hit this time either.
***********************************************
Lindsay Parsons
Grade 9
Mount Saint Joseph Academy
POW for the week of 2/5/96
I began solving this problem of the week by drawing the back yard of your
house. I drew the "ugly" tree in the right hand corner of the yard. Next
I put the veggie garden in on the top edge of the yard. I placed the
willow tree below the other tree on the right side of the yard. Finally I
placed the Lilac bush 708 inches from the "ugly" tree. To find out if when
you cut down the tree it would hit the Lilac bush I made a proportion of
the height of you to the height of the tree, x, over the shadow of you to
the shadow of the tree. Here is the proportion I used...
94 920
70 = x
Next, I set up the equation to find out how tall the tree is in inches.
70 * 920 = 94 x
64400 = 94 x
----- ----
94 94
685.106383 = x
Therefore cutting down this tree will not effect the Lilac bush.
When you are 6 feet or 72 inches tall because of your boots the proportion
is the same except the original 70 inches in the top of the proportion is
changed to 72 so that it looks like this...
94 920
72 = x
The equation I set up for this proportion was...
72 * 920 = 94 x
66240 = 94 x
----- ----
94 94
704.6808511 = x
Also from this equation it appears that the tree that you want to cut down
is not going to effect the Lilac bush.
In either case the Lilac bush won't be hurt with even a scratch because it
is 708 inches away from the tree. So Annie, CHOP AWAY.
***********************************************
Tara Shepherd
Luisa Galdi
Mt. St. Joseph Academy
Grade 10
POW 2/7/96
To figure out if the tree will land on the lilac or not we performed the
following steps:
First find the measure of the angle of the sun's elevation by using the
formula inverse tangent (70/94). YOu will find the measure of the angle
m = 36.67 degrees
Next, using the following formula we figured out the height of the tree
tan 36.67=X/920
X=685 inches or about 57 feet.
Because the tree's height is less than 59 feet, it will not smash the lilac bush.
Considering that in workboots you are close to six feet, we solved the
problem in the same way except we substituted 72 inches for 70 inches in
your height. We found that the tree's height is about 588 feet. This height
is still under 59 feet; however, it will be very close to the lilac bush.
Hopefully, it will not smash your pretty bush.
***********************************************
Annie McIntyre and Liz Croney
Mount Saint Joseph Academy
Grade 9
Pow 2/5-2/9
If you are 5'10 then the tree must be about 57 feet tall.
To find how tall the tree was, we set up a proportion. First
though, we converted everything to inches. We got 920 : 94 : : x :
70. We solved for x and got a number close to 685in. We
converted that to get 57 ft. So, the tree would not hit the
bush.
But considering that you are 6 ft. tall with boots on, we
set up a new proportion- 920 : 94 : : x :72. We found that the
height of the tree is 705 in. which is about 58.7 ft. So the
tree will not hit the bush.
***********************************************
Karl Ljungquist
Grade 9
Cheshire High School
Problem- A tree has to be cut down because it's shading the garden, and it
has to fall between the garden and the willow tree. However, if the tree is
taller than 59, it will smash a lilac bush when it comes down. How do I know
how tall the tree is.
Given Information- At the same time, the tree's shadow was 76'8", and the
persons shadow was 94 inches. The person is in actuality 5'10".
1) Will the tree land on the lilac bush or not?
To figure this out, I thought that the solution must have involved ratios
between the shadow length and the actual length. I only knew both actual and
shadows lengths for one variable, the person. I divided the actual length of
the person (in inches) into the shadow length of the person (in inches).
94
70 = The ratio of actual height to shadow height ( 1: 1.342)
Knowing this new information, I had to figure out the height of the tree with
the ratio.
920 inches 685
1.342 = 685 inches 12= 57 feet.
The tree is 57 feet tall, 2 feet less than the maximum height of 59 feet. The
tree would not crush the lilac bush.
2) Now, consider that in the person's workboots they are 6'. How would the
conclusions be different?
Now I needed a new ratio because of this change in measurement.
94 inches 920 inches(tree)
72 inches = 1: 1.305 1.305 = 704.98 inches.
704.98
12 =58.7 feet. The tree is just under the maximum height of 59 feet.
***********************************************
Paul Laconte
Grade 10
Cheshire High School
Cheshire, CT
To figure out this problem, I decided to put the numbers into a ratio. First
I had to make all the figures the same type( put feet to inches). After all
the figures were in the form of inches, I put them as 70" over 94"( your
actual height to shadow) then I put 920" under "x". So I multiplied 920"
times 70" and then divided by 90". The answer I received was 685". Then I
turned your 59' of space into inches. I got 708". So there is enough room
for you to knock down the tree and not hit your bush.
As soon as you put your workboots on the distance becomes a lot closer.
Before you had 23" of extra space but now you only have 4". To figure out
if the tree would still not hit the bush you just plug 72" into where 70" was
before. You multiply 72" times 920" and divide by 94". This is how I got the
answer of 704" so the tree won't hit the bush but you might be cutting it
close.
Thank you for another difficult problem!
***********************************************
Matt Zibell
Grade 10
Cheshire High School
Cheshire, CT
I first converted all numbers to inches. I then divided 920 by 94 (the
tree's shadow by Annie's shadow). I then took that answer and multiplied it
by 70 (Annie's height in inches). I then divided this answer by 12 (the
number of inches in a foot). The answer I got was 57.09219858. Since the
tree is shorter than 59 feet it will not smash the lilac bush when it is cut
down. If Annie is 6 feet tall in her workboots then the tree would still not
hit the lilac bush. To find this I once again converted all numbers to
inches. I then divided 920 by 94 (the tree's shadow by Annie's shadow). I
then took that answer and multiplied it by 72 (Annie's height in her
workboots). I then divided this number by 12 (the number of inches in a
foot). The answer I got was 58.72340426, this means either way the tree will
not fall on the lilac bush.
***********************************************
Vanessa Fitch
Grade 10
Cheshire High School
Cheshire CT
First we need to convert the measure of the shadow of the tree to inches,
76'8'' to 920'', then your actual height, 5'10'' to 70''. Then we need to
compare the distance like this.
shadows you 94 70 you (actual)
______ = ___ tree (actual)
tree 920 ?
We have to use cross multiplication to find the missing number. 920*70=64400
64400\94=685.1 inches or 57 feet
If the lilac bush is 59 feet away and the tree is 57 feet then you shouldn't
have a problem. However if your workboots make you 6' then you would be 72
inches and it would work out like this.
94 72
_____ = ______
920 ?
920*72=66240
66240\94=704 inches or 58 feet.
In this case I would not cut the tree down because the bush is likely to get
smashed.
***********************************************
Kevin Solli
Grade 10
Cheshire High School
Cheshire CT
Given that you are 5'10'', and your shadow is 94 inches, the tree with a
shadow measuring 76'8'' will not fall on the lilac bush. The tree is only
57'.
I found this by first changing 5'10'' to 70'' and 76'8'' to 920''. Then
knowing that the measure of your shadow is 94'', I found the difference
between your normal height and the height of your shadow. The difference is
24''. I then divided 70 by 24 to find that for every 2.9 inches of actual
height, one inch is added to the height of the shadow. Then I guessed and
checked to find how tall the tree was, knowing that for every 2.9 inches, one
inch of height was added to the shadow. I came up with the conclusion of the
tree being 64'', which is 57'. Since the tree is 57', it will not land on
the lilac bush.
If you are 6' in your workboots, the tree will still not land on the lilac
bush because it is only 58'5''. I found this the same way I found the answer
to the first problem. I found that for every 3.1 inches of actual height,
one inch of shadow is added. I then guessed and checked again to find that
the new height of the tree is only 58'5''.
***********************************************
Rosina Pannone
Grade 10
Cheshire High School
Cheshire, CT
The first thing I did in order to find the height of the tree was to convert
the height of the tree's shadow to inches. That total came out to be 920
inches. Then, I divided 920 inches by 94 inches (the height of your shadow)
to get the ratio of the height of the tree to your height. The ratio came
out to be 9.8:1. I then multiplied by 70 inches (5 feet 10 inches) to get
the height of the tree. This came out to be 57 feet 2 inches. In this case,
no the tree would not hit the lilac bush.
In order to find out the height of the tree using your height as 6 feet in
workboots, I repeated the process but replaced 70 inches with 72 inches (6
feet). In this case, I got the height of the tree to be 58.8 feet, just
barely missing the lilac bush.
***********************************************
Ted Petremont
Tenth
Cheshire, CT
I have to figure out if a tree will fall and hit a lilac bush. To hit a
lilac bush the tree would have to be 59 feet or taller. I was then given the
ratio that a 5'9'' person has a shadow of 94''. That is 1.342857143 shadow
for every 1 foot of height. The tree's shadow is 76'8''. Using 920''/? in
relation to 94''/70''. The question mark is equal to 685.1063829''. The feet
is equal to 57'2'' so the bush is safe where it is.
The same ratio concept is involved with the second question too. It is
920''/? in relation to 94''/72''. Now the question mark represents
704.6568627'' or 58'7''.. Fortunately this also doesn't hit the bush.
***********************************************
Jennifer Rinsland
Grade 10
Cheshire High School
Cheshire, CT
1. In order to determine whether or not the tree is going to fall on the
lilac bush or not, one must determine and convert the actual size of the tree
from the shadow measurements which are given.
It is given that you are 5'10" and that your shadow is 94 inches and that the
shadow of the tree which you are planning to cut down is 76'8". With this
information and the knowledge that the lilac bush which you want to save 59
feet from the tree, one must begin to use mathematical work to figure out if
the tree will land on the bush.
First one must convert the measure of the shadow of the tree from feet to
inches like this:
76*12=912, 912+8=920 so the shadow of the tree is 920 inches
Then one must convert the measure of the height of the person to inches, the
person is 5'10":
12*5=60, 60+10= 70 so the actual height of the person is 70 inches.
Now we must compare the distances like this to determine if the lilac bush
will be smashed by the tree falling on it:
94/920 , 70/
920*70=64400
64400/94=685.1inches
685.1/12=57.09feet.
If the lilac bush is 59 feet away then according to the shadow of the tree
and the ratio of the actual size of the tree, it would be okay if you were to
cut the tree down because you would have 1.91 feet of clearance before you
hit your lilac bush.
2. It is now stated that in your workboots you are measured at 6 feet, so
your shadow will remain the same, but now the ratio between your actual
height and the height of your shadow will be smaller.
To determine the your height in inches, you would do the following:
12*6=72, so you are 72 inches tall
94/920, 72/
920*72=66240
66240/94=704inches
704/12=58.6 feet
In this case the tree would technically not hit the bush but in all actuality
it probably will because there is only 4 inches between where the tree would
fall and where the lilac bush stands if all goes correctly. If anything
happens or an outside element effects the falling of the tree, then the lilac
bush will be smashed. So to be safe it would be better not to remove the tree
and put the garden elsewhere or grow shade flowers because if you cut down
your tree you will probably smash your lilac bush.
***********************************************
Amanda Miller
Grade 10
Cheshire High School
Cheshire, CT
The tree would not hit the lilac bush because using the ratio 9.8:1,
tree:human, the tree would be 57'2". I got that answer first by converting
the tree's shadow of 76'8" into 920" and then dividing 920 by 94 (which is
your shadow in inches), to get the ratio between tree and human. The ratio I
got was 9.8:1. Then I converted 5'10" (your height) to 5.83'. Then I
multiplied 9.8 by 5.83 to get 57.16 which equals 57'2", the height of the
tree.
With you being 6', I used the same ratio, 9.8:1, and got the answer that the
tree would still not hit the lilac bush. The height of the tree would be
58.8'. I got this answer the same way I got the first answer. First, I
converter 76'8" into 920" and then divided by 94, which gave me the same
ratio, 9.8:1. Then, instead of multiplying 9.8 by 5.83, I multiplied by 6.
To get the number 6, I just used the height you said you were closer to with
your workboots on. When I multiplied 9.8 by 6, I got that the tree was 58.8'
which would still leave a little bit of room between the tree and the lilac
bush.
***********************************************
Mary Grelle
Grade 10
Cheshire High School
Cheshire, CT
The first thing I did in this week's Problem of the Week is to convert
the length of the shadow of the tree to inches so that it would match the
units of measurement of the other shadow. I did this by multiplying 5 feet
by 12 because there are 12 inches in a foot. Then I added the other ten
inches. I came up with 70 inches.
Next I found the ratio between 70 inches and the length of the other
shadow (94 inches). I did this by dividing 94 by 70. I rounded the answer
to 1.34. Then I divided the tree's shadow length by 1.34 because the shadows
are longer than the figures are tall, and got 673 inches. I converted this
into feet by dividing by 12 and got 56 feet. Therefore, if the tree was cut
down, it would not hit the lilac bush.
If the other shadow was changed from 5' 10" to 6' then the number of
inches would be 72. When I found the ratio between 94 and 72, by the same
method I used before, I came up with the rounded number of 1.31. Then I
divided 920 by 1.31 and got 702.3 inches. I converted it to feet and got
58.5 feet which still wouldn't hit the bush.
***********************************************
Don Kim
Grade 10
Cheshire High School
Cheshire, CT
Hi, Annie
1) My answer to this problem is that the tree will land safely without
touching the lilac. I solved this problem of converting feet to inches. The
tree's shadow is 920 inches when converted and it gives you that yours was 94
inches. You are 5 feet and 10 inches, which is 70 inches when converted. So
I made an equation set up like this. (920:x=94:70) I multiplied 920 by 70
and divided that number by 94 to get x. and the x turned out to be 685.1063.
Now, I have to divide this number by 12 to get the answer in feet. So after
I did this I got 57.0921 as an answer, which is less than 59 feet. So the
tree will land safely.
2) I used the same method I used in #1 to solve this problem. The equation
in this problem would be (920:x=96:72). You have to remember to adjust your
shadow because of your workboots. Since you got two inches taller with your
workboots, two inches would be added to your shadow.(96) When this equation
is solved, you get 690 as an answer. Finally you divide this number by 12 to
get the answer in feet. The answer is 57.5. So from what I got, the tree
would still be safe to just be landed on the ground. -The End-
Thanks,
Donny Kim
***********************************************
Caryl Anquillare
Grade 10
Cheshire High School
Cheshire, CT
Annie-
Given that you are 5'10", the tree will not land on the lilac. I found this
by converting all measurements into inches. I did that by multiplying any
measurements in feet by 12. Then I added any additional inches leftover to
it. By doing that, you were 70" tall. And the trees' shadow, was 920". So I
divided 94"(your shadows' height) by 70"(your height). That gave me a number
which I rounded off to 1.343. Then I divided 920" (height of trees' shadow)
by 1.343. That gave me about 685". The distance to the lilac is 708" away.
Considering that with your boots, you're about 6', I made that 72" by
multiplying by 12. Then I divided 94" by 72". And I got about 1.306. So I
divided 920" by 1.306. That gave me about 704". So, therefore, it still
shouldn't hit the lilac at 708" away.
-Caryl
***********************************************
Name: Meagan Cihlar
School: Sturgeon Bay High
Grade: 9
Annie-
[\
[ \
x- [ \
[ 70-]\
[________]___\
94
(_____________)
]
920
94 70
--- = --
920 x
x= 57'1"
Lilac bush is safe if you are 5'10".
94 72
--- = --
920 x
x= 58'9"
Lilac bush is safe if you are 6'.
PS-You might want to take out the bush anyway because they only look good for
one month of the year, then they die(at least mine does).
PPS- Sell the tree for firewood--firewood goes for about $120 a cord
now-a-days.
-Meagan
***********************************************
From: Hassan Shamji
Grade: Grade 6
School: River Oaks Public School
Answer: Firstly, I took to mind he was between 5'10" and 6'. So I said he
was about 71 inches. I took the 71 inches and divided it by 94, which was his
shadow
height. The answer to that was .7553191.......let's call that number
"A". So the shadow * "A' is the actual height. The shadow of the
tree is 920 inches, so * that by "A" and you have the height of the tree in
inches. Divide that by 12 to find it out in feet. It is 58 feet. To check this
I assumed his height to be 6'. When you divide 72 inches by 94 you
get .7659574... let's call that "B". Multiply 920 inches by "B" and you get
about
58.7 inches. The bush will be hit because the tree will not fall exactly
near it's stub. It will fall a couple of inches/feet away. This answer depends
on how far up the tree will you cut. I assume you are cutting the tree
from it's base(where it meets the ground).
***********************************************
Using similar triangles the tree is 685.1 in or 57.1 ft. The lilac is
safe!
If you are really 6 ft (72 in) then the tree would be:
72 ?
---- = ----
94 920
? = 704.7 in = 58.7 ft
It still would not flatten the lilac. But it would be very close.
Anna Margush
4th grade
home school
age 9
***********************************************
Bilal Seyal, Fairfield HS, grade 9
By making a proportion, this problem can be solved. Since the person's
height is 5'10" and her shadow is 94 inches, the ratio is 5 5/6:7 5/6 or 35:47.
I'll call the height of the tree x and the shadow of the tree is 76'8". So the
proportion is 35/47 = x/76 2/3. 47x = 2683 1/3, so x is equal to 57 13/141",
which is the height of the tree. The tree is not taller than 59 feet, so
the lilac bush will not be smashed.
Now, we consider the fact that the person is 6' tall. Her shadow is 7 5/6', so
the ratio is 6:7 5/6 or 36:47. The ratio of the height of the tree and its
shadow is still the same, x:76 2/3, so we have to solve the proportion
36/47 = x/76 2/3. 47x is equal to 2760, so x is equal to 58 34/47, which
would be the height of the tree. Since the tree is still less than 59'
tall, the lilac will still not be smashed.
***********************************************
Kyle Halligan, Fairfield HS, grade 9
If you're 5'10" then the tree will fall easily inbounds of the lilac. I
got this by first making a right triangle with 70" as one side and the
other leg as 94". Then I made a similar triangle next to it and had the
bottom leg as 920" and the height as x. I then made a proportion of 94/920
= 70/x. I then solved for x and got 685 5/47". I then divided it by 12
and got 57 13/141' . Since this is under 59', it wouldn't smash the lilac.
If you're 6', then the tree will still not smash the lilac. I got this the
same way as the first one, but I substituted in 72" for 70" because that's
what 6' is in inches. Then after solving for x, I got 704 32/47" and when
that is
represented in feet by dividing by 12, you'd get 58 34/47'. And this is also
under 59', so either way you're safe.
***********************************************
Jason Troske
Grade 10
Cheshire Highschool
Cheshire, Ct
1) I figured that the tree will not hit the lilac bush . To figure this
out I had to do many calculations. First I turned your height from feet to
inches (70"). Next I divided the shadows height (94") by your height (70").
I got 1.3428571". This is what each inch is equal to in the shadows. Now
after determining the difference I turned the trees shadow into inches (920")
and divided the trees inches by the difference (1.3428571) and got 685.1064".
I then turned this into feet by multiplying it by 12. This gave me 57'
which is short enough not to hit the bush.
2) Considering the fact that your boots make you about 6' tall I figured
that the tree will still not hit the lilac bush. First I found that the
difference of the shadows height and the real height is 1.3055555. Next I
turned the trees shadow into inches (704.68087"). I then divided the trees
height by 1.305555 to get 539.75545726". Next I turned that into inches by
dividing it by 12. This gave me 58.723406' which is to short to hit the bush.
***********************************************
Name:Derek Mccarty
School: Sturgeon Bay High
Grade: 9
Annie-
[\
[ \
x- [ \
[ 70-]\
[________]___\
94
(_____________)
]
920
70 94
-- = ---
x 920
x= 57'1"(I rounded up to the higher tenth to absolutely make sure that the
tree won't hit your lilac bush.)
The tree won't hit the bush if you are 5'10".
72 94
-- = ---
x 920
x= 58'8"(I rounded up to the higher tenth to absolutely make sure that the
tree won't hit your lilac bush.)
The tree won't hit your lilac bush when you are in your boots standing 6'.
Now that the ice rink is closed I might try the problem of the month.
-Derek
***********************************************
Lauren Cozzolino
Grade10
Cheshire High School
Cheshire, CT
To figure out if the tree will fit between the veggie garden and the willow
tree without smashing the lilac bush I took into account that the trees
shadow is 76'8" or 920" ( I converted all measurements into inches for
convenience of comparability) Then I looked at the ratio of your shadow
(94") to your height(70"). I divided your height by your shadow to make a
scale (70/94)= .7446808511 Now I have a scale so I can now apply this
knowledge to the tree. .446808511 multiplied by 920" equals 685.106383" To
put this into feet and inches -
57'1". This means that if the lilac bush is 95 feet away and the tree stands
57'1" the lilac bush is safe. Even if you are 5'10" or 72" the tree will not
land on the lilac bush. Here's the math I did:
tree's shadow = 920" - given information
person's shadow = 94"
persons actual height = 72"
scale = 72/94 - divide by actual height by person's shadow
= .7659574468
.7659574468 multiplied by 920" = 704.6808511 or 58'7"
Only if you were 73" tall with a shadow of 94" and the tree's shadow was 920"
would the lilac bush be demolished.
***********************************************
The tree won't fall on the lilac bush either way. The way I figured it
out was to find the ratio of your shadow's height to your actual
height Since you are 5'10", that equals 70 inches. Your shadows height is 94
inches.
I set it up like 70 over 94 equals n over 100. Using this I found the
ratio and applied it to the tree height and the tree shadow. I set it up
like n over 920 (the tree shadow) and 74.4 (the ratio) over 100. I got
686.5 inches, which equals about 57'1" This is under the limit of 59 feet.
Using the 6 feet measurement which equals 72 inches, I put 70 over 94
equals n over 100.
Using the new ratio, I applied to the tree height and the tree shadow,
which would look like n over 920 (tree shadow) equals 76.6 (the new
ratio) over 100. I came up with 704.7 which is 58'8". This is also
under the limit of 59 feet. Both ways, the tree won't fall on the lilac
bush.
Travis Kilpatrick, 8th grade
Art Mabbott
Newport High School
***********************************************
Martin County High School
Stuart, FL
Eric Petersen
Nick Tall pd.8
9th grade
First you need to convert feet into inches. Six feet converts into 72
inches. 76'8" converts into 920 inches. Then we set up a proportion
920":94"=x:72" reducing it to 460":47"=x:72". When we multiplied it we got
the equation 33, 120"=47"x. When dividing the equation we came to the
conclusion that x =704.68085". Converting this back to feet we got
58.723404 feet which rounds off to 59 feet. It doesn't matter if your
wearing your boots or not, either way it won't hit the bush unless you let
it grow a few more inches.
***********************************************
Tim Casale and
Danielle Roberts
grade 10
MCHS
Stuart, Florida
|8')
The answer to this weeks question referring to the height of the tree
is that the tree is 57.09244 feet high and you can rest easy knowing that
your lilac is safe. I came to this conclusion by putting the proportional
sides equal to each other. Thus getting the answer.
5.833/7.833=x/76.667
your height | | ^length of tree's shadow
length of your height of tree
shadow
447.224=7.833x
447.224/7.833=57.09
x=57.09
With the added information, the tree is 58.726 feet in height. It
still will not touch the lilac tree.
6/7.833=x/76.667
460.002=7.833x
/7.833 /7.833
58.726=x
***********************************************
Jaime Uhazie
Martin County High School
Stuart, Fla
POW February 5-9, 1996
To Measure how tall the tree is without your boots (which I don't
think you would be outside without), the first thing that you have to do is
set up similar triangles. Then, convert 76'8" into 920", and 6' into 70".
The, set up a proportion like this: 94/920 = 70/X. Cross multiply this
proportion, and you'll get that 94X = 64400. Divide this by 94 to find
'X', and you'll get 685.106383. Then, divide by 12 to convert the inches
back to feet, and you'll get: 57.09219858, which rounds to around 57 feet.
The tree will not hit the lilac bush.
To Measure how tall the tree is with your boots, the first thing
that you have to do is set up similar triangles. Then, convert 76'8" into
920", and 6'2" into 72". The, set up a proportion like this: 94/920 =
72/X. Cross multiply this proportion, and you'll get that 94X = 66240.
Divide this by 94 to find 'X', and you'll get 704.6808511. Then, divide by
12 to convert the inches back to feet, and you'll get: 58.72340426, which
rounds to around 58'8" feet. The tree will miss the bush, but it is very
risky because there will only be 4" between where the tree falls and the
bush.
***********************************************
Julia Schumm
Edna Evans
Martin County High
Stuart, F.L.
Limber Period 8
The tree is 58.72 feet tall, or 58' 8 2/3" tall. Therefore if you chop
down the tree it will not hit the lilac bush. I obtained this answer by
using a ratio problem. If you are six feet tall and have a 7'10" would
equal the height of the tree (x) over 920" or 76'8" (72"/94"=x"/920"). Or
if your height equals 70" and the height of the shadow of the tree equals
94" then the tree is 57' tall missing the lilac bush with the same formula
as before.
***********************************************
Laura Ejups- 10th
Martin County High School
Stuart Florida
Using the proportion 6:7.83=76.67, I figured out x=58.75. This told me
that the tree would miss the lilac. You can cut the tree when you are
5'10'' and when you are 6'0''. In both cases it won't land on your lilac
bush.
Work:
5.83 x 6.0 x
----- = ---- ------ = -----
7.83 76.67 7.83 76.67
x=57.09 x=58.75
at 5'10''
x= 57.09 / |
at 6'0'' / |
x= 58.75 / |
/ |
/ |
/ |
/ | < TREE
/ |< YOU |
/ | |
/ | 6'0'' OR |
/ | 5'10'' |
------------------------
|-7'10''|
|--------76'8''---------|
***********************************************
Dustin Hicks
Kaitlin Coffinbarger
Grade 9
MCHS
Stuart, FL
First my partner and I converted the shadow and actual measurements into
inches. Those numbers were 94 for your shadow measurement, 920 for the
tree's shadow measurement, and 72 for actual height. Next we set the
following proportion :
Your height Tree Height
Shadow measurement 94 920
--- = ----
Actual measurement 72 answer
So without boots on, yes the lilac bush is o.k. The answer came to be 685
inches. About 57 feet.
With boots we just replaced 72 with 74 and found that the tree is about 2
inches away from 59 feet so hurry up and cut it down.
***********************************************
Celine McElwee & Amy Barbieri
Grade 9
Mount St. Joseph's Academy
To do this problem we decided that we were first going to draw
ourselves a few diagrams to help us figure this out. First we drew a
diagram of you and your shadow and filled in the measurements we knew to
solve for what we didn't know. We then figured that the tangent of the
angle of elevation of the sun(x) was equal to 70" divided by 90". Using
reverse tangent we were able to find that the angle of elevation of the sun
was approximately 37 degrees. Knowing this we plugged it into our diagram
of the tree and its shadow, so we now knew that the shadow is 920" and the
angle of elevation of the sun was approximately 37 degrees by using
tangent. Therefore, the tree is approximately 685 inches or 57'1". If you
were 6' tall, the tree would still be short enough because it would be
approximately 705" or 58'9".
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