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?
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.
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
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!)
To find the volume of the frustum of a pyramid, you need to use the formula:
V = (1/3) * h * (A1 + A2 + √(A1 * A2))
where V is the volume, h is the height of the frustum, A1 is the area of the smaller base, and A2 is the area of the larger base.
In this case, the height of the frustum (h) is given as 12, and the areas of the bases (A1 and A2) are given as 9 and 25, respectively.
Now, substitute these values into the formula to find the volume. Let's calculate step by step:
V = (1/3) * 12 * (9 + 25 + √(9 * 25))
First, calculate the expression inside the square root:
√(9 * 25) = √(225) = 15
Substitute this value back into the formula:
V = (1/3) * 12 * (9 + 25 + 15)
Next, simplify the expression in the parentheses:
V = (1/3) * 12 * (49)
Multiply 12 by 49:
V = (1/3) * 588
Finally, calculate the value of the expression:
V = 196 cubic units
Therefore, the volume of the frustum is 196 cubic units.