Question

a solid oblique pyramid has a regular pentagonal base. the base has an edge length of 2.16 ft and an area of 8 ft². angle acb measures 30°. what is the volume of the pyramid, to the nearest cubic foot? 5 ft³ 9 ft³ 14 ft³ 19 ft³

Explanation:

Step1: Recall volume formula for a pyramid

The volume formula for a pyramid is $V=\frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height.

Step2: Identify base - area and height values

We are given that the base - area $B = 8$ ft². From the right - triangle formed with angle $\angle ACB=30^{\circ}$ and the side adjacent to the angle $BC = 7\sqrt{3}$ ft, we can find the height $h$. Using the tangent function $\tan\theta=\frac{opposite}{adjacent}$, where $\theta = 30^{\circ}$, we have $\tan30^{\circ}=\frac{h}{7\sqrt{3}}$. Since $\tan30^{\circ}=\frac{1}{\sqrt{3}}$, then $\frac{1}{\sqrt{3}}=\frac{h}{7\sqrt{3}}$, and $h = 7$ ft.

Step3: Calculate the volume

Substitute $B = 8$ ft² and $h = 7$ ft into the volume formula $V=\frac{1}{3}Bh$. So $V=\frac{1}{3}\times8\times7=\frac{56}{3}\approx19$ ft³.

Answer:

$19$ ft³