Question

The height of a frustum of a pyramid is 12 and the areas of its bases are 9 and 25. What is the volume of the frustum?

Answer 1

Consider the entire pyramid, including the missing top part. Using similarity, since the top base is 9/25 the area of the bottom, its sides are 3/5 as big.

If the missing portion's height is x, then we have
5/(12+x) = 3/x
x = 18
That makes the height of the entire pyramid 12+18 = 30
So, the whole pyramrid would have a volume of 1/3 * 25 * 30 = 250
The missing part has a volume of 1/3 * 9 * 18 = 54
Thus, the frustrum's volume is 250-54 = 196

Note that since the missing portion is 3/5 as tall as the whole pyramid, its volume is (3/5)^3 of the whole volume. 27/125 * 250 = 54, as above.

Answer 2

make a sketch of a complete pyramid, then cutting off the top pyramid
let the height of the pyramid that was cut off be x units

given : the base of the complete pyramid is 25 square units, so its base side is 5
the base of the cut-off pyramid is 9 square units, so its base is 3

the area of the bases are proportional to the square of the sides
x/(x+12) = 9/25
25x = 9x + 108
x = 6.75

volume of the whole pyramid = (1/3)(base)(height) = (1/3)(25)(18.75) = 156.25 units^3
volume of the cut-off pyramid = (1/3)(base)(height) = (1/3)(9)(6.75) = 20.25

volume of fulcrum is the difference of the two

Answer 3

Scrap my solution and go with oobleck

my equation should have been
x^2/(x+12)^2 = 9/25
which would bring me to the same equation as ooblecks's

(forgot to square the sides, even though I said I should do that!)