Question

Pyramid A is a right square pyramid with a height of 72 m and a square

base with a perimeter of 396 m. Pyramid B has a square base with a
perimeter of 496 m and a height of 239 m. Which pyramid has the
greater volume and by how many times as great?

Answer 1

B by a factor of about 5.208

Explanation:

Given two right square pyramids (A and B) with perimeters 396 m and 496 m, and corresponding heights of 72 m and 239 m, you want to which has the greater volume and by what factor.

Volume

The volume of a right square pyramid is proportional to its height and to the square of its perimeter.

Pyramid B has both a longer perimeter and a greater height, so Pyramid B has the greater volume.

Ratio

The constant of proportionality between the dimensions and the volume is the same for both pyramids, so the ratio of their volumes is ...

Vb/Va = ((Pb)²Hb)/((Pa)²Ha) = (496²·239)/(396²·72) ≈ 5.208

Pyramid B has about 5.2 times the volume of Pyramid A.

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Additional comment

The volume of a right square pyramid is given by ...

V = 1/3s²h . . . . . where s is the side length and h is the height

V = 1/3(P/4)²h = (1/48)P²h . . . . where P is the perimeter of the base