Question

find m.
write your answer in simplest radical form.
miles

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here \(\theta = 30^{\circ}\), the opposite side to \(30^{\circ}\) is \(m\) and the adjacent side is \(11\sqrt{6}\) mi.
\(\tan30^{\circ}=\frac{m}{11\sqrt{6}}\)

Step2: Substitute the value of \(\tan30^{\circ}\)

Since \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\), we have \(\frac{1}{\sqrt{3}}=\frac{m}{11\sqrt{6}}\)
Cross - multiply: \(m=\frac{11\sqrt{6}}{\sqrt{3}}\)

Step3: Simplify the radical

Use the property \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\) (\(a = 6\), \(b = 3\)). So \(m = 11\sqrt{\frac{6}{3}}\)
\(m=11\sqrt{2}\)

Answer:

\(11\sqrt{2}\) miles