Question

find the length x to the nearest whole number.

Explanation:

Step1: Find the length related to the 31 - degree angle

Let the length along the vertical side for the triangle with 31 - degree angle be \(y\). Using the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for the 31 - degree angle, \(\tan31^{\circ}=\frac{x}{y}\), so \(y = \frac{x}{\tan31^{\circ}}\).

Step2: Find the length related to the 62 - degree angle

For the 62 - degree angle, the length along the vertical side is \(y - 360\). And \(\tan62^{\circ}=\frac{x}{y - 360}\), so \(y-360=\frac{x}{\tan62^{\circ}}\).

Step3: Substitute \(y\) from Step1 into Step2

Substitute \(y=\frac{x}{\tan31^{\circ}}\) into \(y - 360=\frac{x}{\tan62^{\circ}}\), we get \(\frac{x}{\tan31^{\circ}}-360=\frac{x}{\tan62^{\circ}}\).

Step4: Solve for \(x\)

\(\frac{x}{\tan31^{\circ}}-\frac{x}{\tan62^{\circ}}=360\). Factor out \(x\): \(x(\frac{1}{\tan31^{\circ}}-\frac{1}{\tan62^{\circ}})=360\).
We know that \(\tan31^{\circ}\approx0.6009\) and \(\tan62^{\circ}\approx1.8807\).
\(\frac{1}{\tan31^{\circ}}\approx1.6643\), \(\frac{1}{\tan62^{\circ}}\approx0.5317\).
\(x(1.6643 - 0.5317)=360\), \(x\times1.1326 = 360\), \(x=\frac{360}{1.1326}\approx318\).

Answer:

318