Question

solve for x.

Explanation:

Step1: Identify the trigonometric ratio

We have a right - triangle with hypotenuse \( c = 16\) m and angle \( \theta=31^{\circ}\). The side \( x\) is adjacent to the angle \( 31^{\circ}\). The cosine of an angle in a right - triangle is defined as \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \( \cos(31^{\circ})=\frac{x}{16}\).

Step2: Solve for \( x\)

To solve for \( x\), we can multiply both sides of the equation by 16: \( x = 16\times\cos(31^{\circ})\).
We know that \( \cos(31^{\circ})\approx0.8572\). Then \( x=16\times0.8572 = 13.7152\approx13.72\) (or we can use a more precise value of \( \cos(31^{\circ})\) depending on the calculator). If we use a calculator to compute \( 16\times\cos(31^{\circ})\):
First, make sure the calculator is in degree mode. \( \cos(31^{\circ})\approx0.8571673007\), then \( 16\times0.8571673007 = 13.71467681\approx13.71\) (or we can round to a reasonable decimal place, for example, to the nearest tenth, \( x\approx13.7\); to the nearest whole number, \( x\approx14\)). But if we follow the standard calculation:

\( x = 16\times\cos(31^{\circ})\approx16\times0.8572 = 13.7152\approx13.7\) (or more accurately, using a calculator for better precision)

Answer:

\( x\approx13.7\) (or \( 13.72\) or \( 14\) depending on the level of precision. If we use a calculator with more precision, \( 16\times\cos(31^{\circ})\approx13.71\))