Question

VOLUME OF A TRUNCATED SQUARE PYRAMID 1. Suppose we know that the volume of a pyramid with square base is a third of the product of the height and the area of the base. We want to derive a formula for the volume of a truncated square pyramid having lower base of side a, upper base of side b, and height h. (a) If we put the top back onto the truncated pyramid so that the restored pyramid has height H, use similar triangles to show that H = ah / (a - b). (b) Now subtract the small upper pyramid from the large one to derive the formula V = 1/3 * h(a^2 + ab + b^2) for the volume of the truncated pyramid. Truncated pyramid Cross section of "restored" pyramid (c) Recall the Egyptian recipe: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top, you are to square this 4, result 16; you are to double 4, result 8; you are to square 2, result 4; you are to add the 16, the 8, the 4, result 28; you are to take a third of 6, result 2; you are to take 28 twice, result 56. See, it is 56. You will find the formula right!" Show that the formula in (b) yields the correct volume for this problem. 2. Derive the volume of the truncated pyramid in Problem #1(b) using the calculus technique known as "volumes by slicing."

          VOLUME OF A TRUNCATED SQUARE PYRAMID
1. Suppose we know that the volume of a pyramid with square base is a third of the product of the height and the area of the base. We want to derive a formula for the volume of a truncated square pyramid having lower base of side a, upper base of side b, and height h.
(a) If we put the top back onto the truncated pyramid so that the restored pyramid has height H, use similar triangles to show that H = ah / (a - b).
(b) Now subtract the small upper pyramid from the large one to derive the formula
V = 1/3 * h(a^2 + ab + b^2)
for the volume of the truncated pyramid.
Truncated pyramid
Cross section of "restored" pyramid
(c) Recall the Egyptian recipe:
"If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top, you are to square this 4, result 16; you are to double 4, result 8; you are to square 2, result 4; you are to add the 16, the 8, the 4, result 28; you are to take a third of 6, result 2; you are to take 28 twice, result 56. See, it is 56. You will find the formula right!"
Show that the formula in (b) yields the correct volume for this problem.
2. Derive the volume of the truncated pyramid in Problem #1(b) using the calculus technique known as "volumes by slicing."

VOLUME OF A TRUNCATED SQUARE PYRAMID
1. Suppose we know that the volume of a pyramid with square base is a third of the product of the height and the area of the base. We want to derive a formula for the volume of a truncated square pyramid having lower base of side a, upper base of side b, and height h.
(a) If we put the top back onto the truncated pyramid so that the restored pyramid has height H, use similar triangles to show that H = ah / (a - b).
(b) Now subtract the small upper pyramid from the large one to derive the formula
V = 1/3 * h(a^2 + ab + b^2)
for the volume of the truncated pyramid.
Truncated pyramid
Cross section of "restored" pyramid
(c) Recall the Egyptian recipe:
"If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top, you are to square this 4, result 16; you are to double 4, result 8; you are to square 2, result 4; you are to add the 16, the 8, the 4, result 28; you are to take a third of 6, result 2; you are to take 28 twice, result 56. See, it is 56. You will find the formula right!"
Show that the formula in (b) yields the correct volume for this problem.
2. Derive the volume of the truncated pyramid in Problem #1(b) using the calculus technique known as "volumes by slicing."

Added by Michael W.

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