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: Find the height of the pyramid

In triangle \(ACB\), \(\tan(30^{\circ})=\frac{\text{height}}{7\sqrt{3}}\). We know that \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), so \(\frac{1}{\sqrt{3}}=\frac{\text{height}}{7\sqrt{3}}\). Solving for height, we multiply both sides by \(7\sqrt{3}\), getting height \( = 7\).

Step2: Calculate the volume of the pyramid

The formula for the volume of a pyramid is \(V=\frac{1}{3}Bh\), where \(B\) is the area of the base and \(h\) is the height. We know \(B = 8\space\text{ft}^2\) and \(h = 7\space\text{ft}\). Plugging these values into the formula, we get \(V=\frac{1}{3}\times8\times7=\frac{56}{3}\approx18.67\). Rounding to the nearest cubic foot, we get \(19\space\text{ft}^3\).

Answer:

\(19\space\text{ft}^3\) (corresponding to the option with \(19\space\text{ft}^3\))