Question

given a square pyramid with a volume of 180 cubic meters and a height of 12 meters, which of the following expressions correctly solves for the side length of the square base, s?
$\bigcirc s=\sqrt{\frac{(3)(180)}{12}}$
$\bigcirc s=\sqrt{\frac{(12)(180)}{3}}$
$\bigcirc s=\sqrt{\frac{12}{3(180)}}$
$\bigcirc s=\sqrt{\frac{180}{(3)(12)}}$

Explanation:

Step1: Recall the volume formula for a square pyramid

The volume formula for a square pyramid is \( V=\frac{1}{3}s^{2}h\), where \(V\) is the volume, \(s\) is the side - length of the square base, and \(h\) is the height.
We are given that \(V = 180\) cubic meters and \(h=12\) meters. Substitute these values into the formula: \(180=\frac{1}{3}s^{2}(12)\).

Step2: Solve the equation for \(s^{2}\)

First, simplify the right - hand side of the equation \(180=\frac{1}{3}s^{2}(12)\). \(\frac{1}{3}\times12 = 4\), so the equation becomes \(180 = 4s^{2}\). Then, solve for \(s^{2}\): \(s^{2}=\frac{180\times3}{12}\) (multiply both sides by 3 to get \(180\times3=12s^{2}\), then divide both sides by 12).

Step3: Solve for \(s\)

Since \(s=\sqrt{s^{2}}\), and \(s^{2}=\frac{(3)(180)}{12}\), then \(s=\sqrt{\frac{(3)(180)}{12}}\)

Answer:

\(s=\sqrt{\frac{(3)(180)}{12}}\)