Question

find c.
write your answer in simplest radical form.
centimeters

Explanation:

Step1: Identify the trigonometric ratio

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), adjacent side \(c\), and hypotenuse \(4\sqrt{6}\) cm.
So, \(\cos30^{\circ}=\frac{c}{4\sqrt{6}}\)

Step2: Substitute the value of \(\cos30^{\circ}\)

We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\).
Substituting in the above equation: \(\frac{\sqrt{3}}{2}=\frac{c}{4\sqrt{6}}\)

Step3: Solve for \(c\)

Cross - multiply: \(c = 4\sqrt{6}\times\frac{\sqrt{3}}{2}\)
Simplify: \(c = 2\sqrt{18}\)
Further simplify \(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\)
So, \(c = 2\times3\sqrt{2}=6\sqrt{2}\)

Answer:

\(6\sqrt{2}\) centimeters