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 cub foot? 5 ft³ 9 ft³ 14 ft³ 19 ft³

Explanation:

Step1: Find the height of the pyramid

In right - triangle \(ABC\), we know that \(\tan\angle ACB=\frac{AB}{BC}\). Given \(\angle ACB = 30^{\circ}\) and \(BC = 7\sqrt{3}\text{ ft}\).
Since \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\), and \(\tan\angle ACB=\frac{AB}{BC}\), then \(AB = BC\times\tan30^{\circ}\).
Substitute \(BC = 7\sqrt{3}\) into the formula: \(AB=7\sqrt{3}\times\frac{1}{\sqrt{3}} = 7\text{ ft}\).

Step2: Calculate the volume of the pyramid

The volume formula for a pyramid is \(V=\frac{1}{3}Bh\), where \(B\) is the base area and \(h\) is the height.
Given \(B = 8\text{ ft}^2\) and \(h=7\text{ ft}\).
Substitute the values into the formula: \(V=\frac{1}{3}\times8\times7=\frac{56}{3}\approx19\text{ ft}^3\).

Answer:

\(19\text{ ft}^3\)