Geometry help?

Question

One can of pumpkin pie mix will make a pie of diameter 8in. If two cans of pie mix are used to make a larger pie of the same thickness, find the diameter of that pie. Use square root of 2 = 1.414.

Answer 1

The area is doubled, so the diameter is increased by a factor of sqrt(2).

New diameter = 8*sqrt(2) = 8*1.414 = 11.312 inches

Answer 2

The area of a circle is A = pi * R * R = pi * R^2, where R is the radius of a circle and ^ is the exponentiation operator.

The area of the one-can pie is A_1 = pi * (8 / 2)^2 = pi * 16.

Assuming that the edge of the pie is vertical, the volume of the mix is V_1 = A_1 * D, where D is the depth or thickness. In practice, the edge if the pie is sloped so this equation is only an approximation. Since we are given no information about the angle of the slope, the assumption is about the best we can do.

The volume of the two-can pie is V_2 = 2 * V_1 = 2 * A_1 * D = 2 * pi * 16 * D.

We want V_2 = pi * R_2^2 * D. This is equation is equal to the one above:

V_2 = 2 * pi * 16 * D = pi * R_2^2 * D. Solve for R_2 and the result is 4 * sqrt(2). We want the diameter or twice this radius.

The answer is 8 * sqrt(2) inches.

Intuitively, this is the expected answer once you get a feel for this kind of problem. Did I just do you homework?

Answer 3

The area of a circle grows like the square of the diameter. So, if a circle's diameter is multiplied by 2, its area will be multiplied by 4.

The volume of a cylinder is the height multiplied by the area of the circle.

A pie is (very close to) a cylinder. If you are given sufficient filling to double the volume of the pie, then that means that the area of the pie will be doubled.

That means that the diameter of the pie will be multiplied by the square root of 2 (√2)
Because (√2) squared = 2.

Therefore your bigger pie will have a diameter of 8 times the square root of 2 =
8 * 1.414 = 11.312 inches.

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If you wanted to make a pie with a diameter of 8*2 = 16 inches, you would need (2^2) cans (four cans)

If you wanted to make a pie with a diameter of 8*3 = 24 inches, you'd need 3^2 = 9 cans.

If you wanted to make a pie with a diameter of 8*10 = 80 inches, you'd need 10^2 = 100 cans and a very large oven.

Answer 4

set up a proportion to compare the areas of the pie surfaces to each other.
A=pi r^2

1/16pi = 2/x
cross multiply

x = 32pi

32 pi is the area of the second pie. to find the radius, just plug it into the area equation.

32pi = pi r^2
divide by pi on both sides to solve for r

32 = r^2

sqrt 32 = r

sqrt 32 can be broken into sqrt4*sqrt4*sqrt2
sqrt 32 = 2*2*sqrt2

r = 4sqrt2

d = 8 sqrt2 and since sqrt2 = 1.414
d = 11.312 in.

Answer 5

use the ratio and proportion:
1/8 = 2/X
X = 16