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I have enjoyed reading your solutions to my shotgun target problems, so I
have another one for you.
I'm constructing another target for long-range shooting. The target will have a six-inch diameter black bull's-eye, and concentric circles with diameters of 12", 18", and 24" respectively, to form alternating white and black bands. There is also a 24" square circumscribed about the outermost circle. Got it? Okay. What is the area of the bull's-eye and what is the area of each of the white and black bands making up the target? (There are four areas to determine - the bull, two white bands, and one black band.) While we're at it, what's the area between the 24" outer circle and the 24" square? What do you notice about the relation between the area of each section and the band just outside it? EXTRA: Now that you're warmed up, can you identify WHY the relation in the last question exists? (You might want to try some similar setups with targets of different sizes to see if that helps.) SolutionsFinding the areas this week wasn't too much of a problem for most of you, though some of you didn't read carefully and you found the area of all the circles, not just the bands. 78 folks got it right, and 64 got it wrong. Most of those who got it wrong neglected to include a statement about the relation between consecutive regions. I wasn't looking for anything fancy with that, I just wanted you to notice that there was a constant increase in area as you went from the bull's-eye to the outer ring. This could take many different forms. The most common ones were that the difference is 18pi, or 56.5, or that the difference was equivalent to two bull's-eyes - that's a good observation! Some people pointed out that the areas were odd multiples of 9pi - 1, 3, 5, 7. It's good to be able to see patterns like that in your work. The bonus was to try to figure out why this relation existed. A number of people took a stab at it. I have included two of the solutions below - those of Jenny Kaplan of Castilleja Middle School and Jason Chiu of Laramie Junior High School. Give them a read.
When doing a problem like the bonus, it's a good idea to throw in some
x's instead of numbers and see if you can get it to work out
algebraically. For example, represent the area of one ring as The next time that I use a problem like this (maybe it will be next year), I might not give credit to people who use "pie" instead of "pi." It's a simple thing, and it shows that you have some understanding of what pi is - it's a symbol that looks sort of like this - TT - that is a letter in another alphabet (Greek), and since the TT symbol is hard to write sometimes, we spell the word out by saying pi. "Pi" stands for a particular number, and it's important that you use it correctly. A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.
From: Jason Chiu
Grade: 9
School: Laramie Junior High School, Laramie, Wyoming
Using the formula A=pi*r^2, the area of each circle can be found. The
radius of the circles are 3, 6, 9, and 12 inches, which makes their areas 9*pi,
36*pi, 81*pi, and 144*pi, respectively. The area of the bull's-eye is 9*pi
(inner circle); the first white ring (inner) is pi(36-9)=27*pi; the black ring
is pi(81-36)=45*pi; and the outer white ring is pi(144-81)=63*pi.
The area of the 24 inch to a side square is 24^2=576, and that minus the
area of the largest circle is 576-144*pi.
The area increases proportionally to the increase of the radius of the
circle. Because the increase of the radius is constant in this case, the
pattern is constant.
I will have a mathematical proof to answer this part of the problem. I
will prove that it is linearly increasing if c is. Let the inner circle (of the
two) have radius x and the increase be a constant, c. So the area of the band
is pi((x+c)^2-x^2), which simplifies to pi(2cx+c^2). It is linear because the
degree of the only x is 1. It is increasing linearly and if that is, then the
relationship happens.
From: Jenny Kaplan
Grade: 7
School: Castilleja Middle School, Palo Alto, California
I gave each of the different bands or circle a number.
Starting with the bulls eye as 1, each of the next bands
had the next numbers. The area of number 1 (or the bulls-
eye) is 9pi. The area of number 2 is 27pi. The area of
number 3 is 45pi, and the area of number 4 is 63pi. Each
of these numbers was found from taking band that you want
to find the area’s of area and subtracting the area of the
circles inside it.
The area between the outer square and the circle inside it
is 576 - 144pi. This was found by subtracting the area of
the circle from the area of the square.
For this next question I thought of everything in terms of
a multiple of 9. Therefore the area of each circle was 9*1pi,
9*4pi, and 9*9pi. The areas of the corresponding bands are
9*3pi, 9*5pi, and 9*7pi. I have noticed 2 things. The first
one is that all of the circles are 9 times each consecutive
square. The second thing is that each of the bands have a
difference of 18pi between them (as well has having 9 times the
next consecutive odd numbers).
EXTRA:
First I looked at the pattern with the circles, and found out
why they are 9 times each consecutive square number. All of the
radii of the circles given are divisible by 3. To then find the
area you square the 3 as well as the number that 3 was multiplied
by. If you were to take out all of the 9s then you would be left
with the squares of all of the other numbers. They happen to be
consecutive squares because each of the radii were either 3*1,
3*2, 3*3, or 3*4.
For the pattern of the bands this is what I did: If you look at
the pattern, you will see that you skip all of the even multiples
of 9. Since there is a difference of 9 between each power, there
is a difference of 18 between all of the odd multiples of 9 in this
pattern.
The following students submitted correct solutions this week:Mark Yang, Grade 7, Alki Middle School, Vancouver, WashingtonTiffanie Lam, Grade 8, Sequoia Middle School, Pleasant Hill, California Aaron Mertz, Grade 8, Plum Grove Junior High School, Rolling Meadows, Illinois Gordon Bockus Jr., Grade Freshman, Eastern Oklahoma State College, Wilburton, Oklahoma Natasha Fedner, Grade 9, Bexley High School, Bexley, Ohio Jessi Kennedy, Grade 9, Amador Valley High School, Pleasanton, California Michelle Hrdina, Grade 10, South Kitsap High School, Port Orchard, Washington Joey Fenton, Grade 10, South Kitsap High School, Port Orchard, Washington Aaron Ogletree, Grade 10, South Kitsap High School, Port Orchard, Washington Glen Melnik, Grade Teacher, Piedmont Independent Learning High School, Oakland, California Katsu Sato, Grade 6, Troy Intermediate School, Avon Lake, Ohio Catherine Mangasi, Grade 12, Wilburton High School, Wilburton, Oklahoma Curt Henn and Matt Henn, Grade 9, Bexley High School, Bexley, Ohio Josh Boyette, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Abby Lindsay , Grade 6, Troy Intermediate School, Avon Lake, Ohio Dan and Steve, Grade 6, Troy Intermediate School, Avon Lake, Ohio Amber Jensen, Grade 5, Crystal Springs Elementary Arielle Cohen, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Lauren Brown, Grade 9, Hudson's Bay High School, Vancouver, Washington Sarah Hesson, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania DeDe DiDonato, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania Joseph Bovard, Grade 9, Hudson's Bay High School, Vancouver, Washington Chris Lauber, Grade 9, Smoky Hill High School, Aurora, Colorado Kim Harris, Grade 8, Albright Middle School, Houston, Texas Crystal Cannaday, Grade 9, Franklin County High School, Rocky Mount, Virginia Jason Shertzer and Karl and Sean and Jeff, Grade 9, Skyview High School, Vancouver, Washington Jonathan Emmons and Amey Adkins, Grade 9, Franklin County High School, Rocky Mount, Virginia Melanie, Grade 9, Skyview High School, Vancouver, Washington Jacob Casey, Grade 9, Skyview High School, Vancouver, Washington Victor Hsieh, Grade 9, Hudson's Bay High School, Vancouver, Washington Isaiah Wyckoff, Grade 9, Skyview High School, Vancouver, Washington Jessica Barclay-Strobel, Grade 10, Oak Park and River Forest High School, Oak Park, Illinois Avrum Tilman, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Chester Chan, Grade 6, Odle Middle School, Bellevue, Washington Kristin Loyd, Grade 8, Discovery Middle School, Vancouver, Washington Jia-Xin Wang, Grade 9, American Embassy School, New Delhi, India Hannah Margoles, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Nicky Kim, Grade 10, American Embassy School, New Delhi, India Jason Chiu, Grade 9, Laramie Junior High School, Laramie, Wyoming Alex Chen, Grade 7, Odle Middle School, Bellevue, Washington Jenny Kaplan, Grade 7, Castilleja Middle School, Palo Alto, California Ryan Bates, Grade 9, Skyview High School, Vancouver, Washington Mary Park, Grade 8, Odle Middle School, Bellevue, Washington Alessandro Binotti, Grade , American Embassy School, New Delhi, India Kristen Lee, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Tova Gardner and Brianna Sterling, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Maja White, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Clayton, Grade , Redmond High School, Redmond, Oregon Peter Lindgren, Grade , Farila Skola, Farila, Sweden C. G., Grade 9, Germantown Academy, Fort Washington, Pennsylvania Emily Castor, Grade 9, Granada High School, Livermore, California Lyrica Hubbard, Grade , Redmond High School, Redmond, Oregon Alison Miller, Grade 6, homeschooled, Niskayuna, New York Chaim Bloom, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Sarah Kress, Grade 9, Mount Saint Joseph Academy, Flourtown, Pennsylvania Kevin Palmer, Grade 9, Granada High School, Livermore, California Sundhar Ramalingam, Grade 9, Southeast Raleigh High School, Raleigh, North Carolina Tanya Colburn, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania Anna Warszawa, Grade 9, Germantown Academy, Fort Washington, Pennsylvania Laura Roos, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Alex Chernyavsky, Grade , Akiba Hebrew Academy, Merion, Pennsylvania Abin Bandy, Grade , Germantown Academy, Fort Washington, Pennsylvania Abin Bandy, Grade 9, Germantown Academy, Fort Washington, Pennsylvania Thomas Kuo, Grade 10, Burroughs High School, Ridgecrest, California Jennifer Liang, Grade 8, Odle Middle School, Bellevue, Washington Niki Weber, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Lauren Rossi and Anne Hines, Grade , Germantown Academy, Fort Washington, Pennsylvania Catalina Anghel, Grade 11, Mackenzie High School, Deep River, Ontario, Canada Teneal Dollar, Grade 10, Shelby County High School, Columbiana, Alabama Jackie Garner, Grade 11, Shelby County High School, Columbiana, Alabama Erika Mitchell, Grade 11, Shelby County High Schoo, Columbiana, Alabama Katy Crumpton, Grade 10, Shelby County High School, Columbiana, Alabama Julie Dennis, Grade 10, Shelby County High School, Columbiana, Alabama Christy Thornburg, Grade 10, Shelby County High School, Columbiana, Alabama Frannie Laks, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Alex Morgovsky, Grade , Akiba Hebrew Academy, Merion, Pennsylvania Lia LaBrant, Grade 8, Discovery Middle School, Vancouver, Washington Zimran Douglas, Grade 12, Wingate High School, Brooklyn, New York |
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