Question

find s. write your answer in simplest radical form. feet

Explanation:

Step1: Recall the property of a 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, if the side adjacent to the 30° angle is \(x\), and the hypotenuse is \(h\), then \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), the adjacent side \(=15\sqrt{6}\) and the hypotenuse \(=s\).
We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\). So, \(\cos30^{\circ}=\frac{15\sqrt{6}}{s}\).

Step2: Solve for \(s\)

Since \(\frac{\sqrt{3}}{2}=\frac{15\sqrt{6}}{s}\), we can cross - multiply. We get \(s\times\sqrt{3}=2\times15\sqrt{6}\). Then \(s = \frac{30\sqrt{6}}{\sqrt{3}}\).
Using the rule \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\) (\(a = 6\), \(b = 3\)), we have \(s=30\sqrt{\frac{6}{3}}\).
Since \(\frac{6}{3}=2\), then \(s = 30\sqrt{2}\).

Answer:

\(30\sqrt{2}\)