Question

solve for x. round to the nearest tenth, if necessary.

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle \( \triangle MLK\) with \( \angle L = 90^{\circ}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 31^{\circ}\), the opposite side to \(31^{\circ}\) is \(LK\), and the hypotenuse is \(MK = 96\), and the adjacent side to \(31^{\circ}\) is \(ML=x\). We use the cosine ratio: \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Substitute the values into the cosine formula

We know that \(\cos M=\frac{ML}{MK}\). Given \(M = 31^{\circ}\) and \(MK = 96\), so \(\cos(31^{\circ})=\frac{x}{96}\).

Step3: Solve for \(x\)

Multiply both sides of the equation by \(96\): \(x = 96\times\cos(31^{\circ})\).
Since \(\cos(31^{\circ})\approx0.8572\), then \(x=96\times0.8572\).
\(x = 82.3\)

Answer:

\(x\approx82.3\)