SOLUTION: Find the volume of a pyramid with height of 15 feet and whose base is a square with edges of lengths 10 feet. Show your work using a definite integral. Check your answer using the

Algebra ->  Equations -> SOLUTION: Find the volume of a pyramid with height of 15 feet and whose base is a square with edges of lengths 10 feet. Show your work using a definite integral. Check your answer using the      Log On


   

Question 1142808: Find the volume of a pyramid with height of 15 feet and whose base is a square with edges of lengths 10 feet. Show your work using a definite integral. Check your answer using the formula for the volume of a pyramid.
Answer by greenestamps(13198) About Me  (Show Source):
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We will integrate in the direction of the height; our "thin slices" of the pyramid will be squares whose side length is in constant proportion with the height.

The side of the base is 10 and the height is 15. That means every cross section will have a side length that is 2/3 of the height at that point. So if h is the height,

B = ((2/3)h)^2 = (4/9)h^2

Integrate from 0 to 15:
Integral of ((4/9)h^2dh) = (4/27)h^3 (+C)

The integral evaluated from 0 to 15 is

(4/27)(15^3) = 4(5^3) = 500

The volume of the pyramid by calculus is 500 cubic feet.

By the formula for the volume of a pyramid:

(1/3)(B)(h) = (1/3)(100)(15) = 500