Question

find ac.
write your answer in simplified, rationalized form. do not round.
ac =

Explanation:

Step1: Identify the triangle type

This is a right - triangle with a \(30^{\circ}\) angle at \(A\), right - angled at \(C\) (assuming the right angle is at the vertex with the right - angle symbol). The hypotenuse \(AB = 16\sqrt{10}\). In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\) and the adjacent side to \(\theta\) is \(AC\), and the hypotenuse is \(AB\).

Step2: Recall the cosine formula

We know that \(\cos(30^{\circ})=\frac{AC}{AB}\). We know that \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\) and \(AB = 16\sqrt{10}\).

Step3: Solve for \(AC\)

Substitute the values into the formula: \(AC=AB\times\cos(30^{\circ})\).
\(AC = 16\sqrt{10}\times\frac{\sqrt{3}}{2}\)
Simplify the expression: \(16\div2 = 8\), so \(AC = 8\sqrt{10\times3}=8\sqrt{30}\)

Answer:

\(8\sqrt{30}\)