Solutions
Annie says:
This week's statistics are pretty good, lots of correct answers and only a few incorrect, showing that not everyone was affected by the snow. A couple of the incorrect solutions were due to units. As Benjamin Warfield said, "All your units are screwy," and indeed they are. But they are the units that we are used to seeing in such a situation - a long distance is given in miles, a not-so-long distance in yards, and a small distance (like snow) in inches.
A couple of people sent in supplemental solutions after being fooled by the units. Many people translated into feet or even inches! I like yards because the numbers aren't too bad, and though you have to use 5/6 for the snowfall, that works out okay.
They have 40 miles of snow to plow at the airport - that's a lot of pyramids if you let this guy do the job, eh?
Below are some highlights. The names of the people who submitted correct solutions and most of the solutions are also available, including some from our "adult" crowd (when you graduate from high school you automatically become an adult in this arena), and our first solutions from Maine and the Philippines.
Submitted by Jim Hamalainen
Step 1. Convert all lengths or amounts into inches or cubic inches.
1 mile = 63360 inches and 50 yrds.= 1800 inches
Step 2. Find total amount of snow collected by plow.
(length times width times height)
63360" x 1800" x 30" = 3,421,440,000 cubic inches of
snow.
Step 3. Divide snow volume into four even amounts (for 4 pyramids)
= 855,360,000 cubic inches of snow per pyramid
Step 4. To find height, assume that there is snow filling in space
above the present pyramid in such a way to form a rectangular
prism. The volume of this "box" would be three times the
amount of one pyramid, or 2,566,080,000 cubic inches of snow.
Step 5. Divide the volume of the "box" by the base area of the
pyramid (which is 1800" x 1800"= 3,240,000 square inches)
to find the height of the pyramid.
The final answer is 22 yds tall (or 66 ft or 792 inches tall).
Timothy Farrell
Grade: Sophomore (10)
School: Bishop Walsh Middle/High School
Answer: 50 yards x 1 mile x 30 inches =
150 feet x 5280 feet x 2.5 feet =
1980000 feet^3
He made four pyramids, each exactly the same, so we'll just
take 1/4 of the above height to simplify the situation.
.25x1980000 feet^3 = 495000 feet^3
Pyramid volume formula: area of base x height/3
area of base = 50 yd ^2 =
150 feet x 150 feet =
22500 feet^2
Volume of pyramid:
Let n = height of each pyramid
22500 feet^2 x (n feet)/3 = 495000 feet^3
22500 feet^2 x (n feet)/3x3 = 495000 feet^3x3
22500 feet^2 x (n feet) = 1485000 feet^3
22500 feet^2 / 22500 feet^2 x (n feet) = 1485000 feet^3/22500 feet ^2
(n feet) = 66 feet
Therefore, the size of the pyramid is:
50 yards wide x 50 yards long x 66 feet (22 yards) tall.
Cassie Gorish
Grade: 8
School: Murray Junior High
First, I went into the Smithsonian Institute on the Internet and
looked up the Great Pyramid of Giza. Using this information,
I could make a proportion, but I chose another way. With my dad,
I looked up in his Mechanical Engineer Manual the equation for
finding the volume of pyramids. By finding the volume of the
snow the man plowed and dividing it into four, I can find the
volume of each pyramid: I found the volume to be 495,000 cubic
feet.
The equation for finding the volume of pyramids is:
V = 1/3 height X area of the base
Since I know the volume and the base length, I can solve for
height:
495000 = 1/3 h X 22500
22 = 1/3 h
66 = height
So, assuming that he built all the pyramids of equal dimensions,
each pyramid is 66 feet in height.
Katie Walder
Grade 9
Mount Saint Joseph Academy, Flourtown, PA
This problem seemed very complicated at the beginning, but I
think I figured it out.
The area the guy was plowing was a 3-dimensional rectangle.
I labeled its length as a mile, or 5,280 feet. I labeled its width
as 50 yards, or 150 feet. I labeled its height as 30 inches, or
2.5 feet.
I then plugged all my numbers into the formula for the volume of a
rectangular prism, as follows:
V = (l) x (w) x (h)
V = (5280) x (150) x (2.5)
V = 1,980,000 cubic feet
This is the volume of all the snow, so I divided it by 4 to get
the volume of one pyramid, which equalled 495,000. I then used
the formula for a pyramid, which is as follows:
V = 1/3 Bh
495000 = 1/3 (150 x 150)h
66 = h
Each pyramid must then have a height which equals 66 feet, or 22
yards.
Benjamin Warfield
Grade 12
School: Garfield High School, Seattle
All your units are screwy. Nonetheless, using the formula
V=1/3*b*h and a whale of a lot of conversion factors, I find
that the volume of each pyramid is 220,000 cubic yards, and that
the height (in the one English unit you didn't use) is 66 feet.
Brian Gordon
Dartmouth '92
First, the volume of the snow is 5280 feet x 150 feet x 2.5 feet
= 1,980,000 cubic feet of snow.
Next, this has to be equal to four times the area of each pyramid.
Each pyramid has volume 1/3 x 150 feet x 150 feet x height.
So the equation is:
4 * 1/3 * 150 * 150 * h = 1,980,000
Divide by everything to get h = 66 feet, or 22 yards.
--bri
P.S. We only got about 18-20 inches of snow here in Middletown, CT.
Ken Duisenberg
Grade: Post-Master's
School: California State University, Chico
The pyramids are each 22 yards high.
The area of a pyramid, with base area A and height h, is:
Ah/3.
The base area of each of the four pyramids is 50*50 sq.yds.
The total volume of the pyramids is thus 50*50*4h/3 cu.yds.
The total volume of snow (assuming it keeps the same volume
upon being plowed - quite unlikely, I'd think) is (1mi. = 1760 yd.):
50*1760*(5/6) cu.yds.
Setting these equal and solving for h, finds h = 22yds.
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