Question

the two hexagonal pyramids shown below are similar. the bases of both pyramids are regular hexagons. the height of the smaller pyramid is \\( \frac { 1 } { 4 } \\) the height of the larger pyramid. the length of a side of the base of the smaller pyramid is 1 inch, and its height is 2 inches. what is the volume, v, of the larger pyramid? (note: the apothem of a regular hexagon is \\( \frac { \sqrt { 3 } } { 2 } \\) times the length of one side.) 1 of 5 questions \\( v = 1 2 8 \sqrt { 3 } \\) cu in \\( v = 6 4 \\) cu in \\( v = 6 4 \sqrt { 3 } \\) cu in \\( v = 1 2 8 \\) cu in

Explanation:

Step1: Find the height of the larger pyramid

Given the height of the smaller pyramid \(h_{s}=2\) inches and \(h_{s}=\frac{1}{4}h_{l}\) (where \(h_{l}\) is the height of the larger pyramid).

$$h_{l}=4h_{s}$$

$$h_{l}=4\times2 = 8$$

inches.

Step2: Find the side - length of the base of the larger pyramid

Since the pyramids are similar, and the side - length of the base of the smaller pyramid \(s_{s}=1\) inch. The ratio of side - lengths is the same as the ratio of heights. Let \(s_{l}\) be the side - length of the base of the larger pyramid. \(\frac{s_{s}}{s_{l}}=\frac{h_{s}}{h_{l}}\), so \(s_{l}=4s_{s}=4\) inches.

Step3: Calculate the area of the base of the larger pyramid

The formula for the area of a regular hexagon \(A=\frac{1}{2}\times perimeter\times apothem\).
The perimeter of the base of the larger pyramid \(P = 6s_{l}=6\times4 = 24\) inches.
The apothem \(a=\frac{\sqrt{3}}{2}s_{l}\), substituting \(s_{l} = 4\), we get \(a=\frac{\sqrt{3}}{2}\times4=2\sqrt{3}\) inches.

$$A=\frac{1}{2}\times24\times2\sqrt{3}=24\sqrt{3}$$

square inches.

Step4: Calculate the volume of the larger pyramid

The formula for the volume of a pyramid \(V=\frac{1}{3}Ah\) (where \(A\) is the base area and \(h\) is the height).
Substituting \(A = 24\sqrt{3}\) and \(h=h_{l}=8\)

$$V=\frac{1}{3}\times24\sqrt{3}\times8$$

$$V = 64\sqrt{3}$$

cubic inches.

Answer:

C. \(V = 64\sqrt{3}\text{ cu in}\)