Question

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Sarah has two similar regular pyramids with pentagon-shaped bases. The smaller has a scale factor of 2:3 when compared to the larger. Only the smaller pyramid is shown.

She calculates the area of the base of the pyramid (through long, hard work) to be 220.22 square units. The height of the pyramid is 15 units. Now she needs to calculate the volume of the pyramid.

(a) Calculate the volume of the pyramid for Sarah.

(b) "Oh, no!" Sarah exclaims. "Now I have to go through all this hard work again to find the volume of the larger pyramid!" Does she? Explain.

(c) Calculate the volume of the larger pyramid for Sarah.

Answer 1

a) The volume of the pyramid 440.44 cube units

(b) No she doesn't

(c) The volume of the larger pyramid is 1,486.485 cube units

Explanation:

The given parameters are;

The scale factor of the pyramids, S.F. = 2:3

The base area of the small pyramid, = 110.11 square units

The height of the small pyramid, h₁ = 12 units

(a) The volume of a pyramid, V = (1/3) × Area of base × The height of the pyramid

Therefore;

The volume of the small pyramid, V₁ = (1/3) × 110.11 square units × 12 units

V₁ = 440.44 cube units

The volume of the small pyramid, V₁ = 440.44 cube units

(b) No she does not have to go through all the hard work again to find the volume of the larger pyramid

She only has to make use of the scale factor relationships of the two pyramid to calculate the volume of the larger pyramid

The volume scale factor = (The linear scale factor)³

(c) The linear scale factor of the pyramids = 2:3 = 2/3

Therefore;

The volume scale factor of the pyramids = (2/3)³ = 8/27

To find the volume of the larger pyramid, V₂, from the volume of the smaller pyramid, V₁, we multiply the volume of the smaller pyramid, V₁, by 27/8 as follows;

V₂ = V₁ × (27/8)

Therefore;

The volume of the larger pyramid, V₂ = 440.44 cube units × (27/8) = 1,486.485 cube units

The volume of the larger pyramid, V₂ = 1,486.485 cube units.