Question

find h as indicated in the figure. h = (round to the nearest integer as needed.)

Explanation:

Step1: Use tangent function for both angles

Let the two - part lengths of the base be \(x\) and \(y\) such that \(x + y=349\). We know that \(\tan(25.7^{\circ})=\frac{h}{x}\), so \(x = \frac{h}{\tan(25.7^{\circ})}\), and \(\tan(40.4^{\circ})=\frac{h}{y}\), so \(y=\frac{h}{\tan(40.4^{\circ})}\).

Step2: Substitute \(x\) and \(y\) into \(x + y = 349\)

\(\frac{h}{\tan(25.7^{\circ})}+\frac{h}{\tan(40.4^{\circ})}=349\). Factor out \(h\): \(h(\frac{1}{\tan(25.7^{\circ})}+\frac{1}{\tan(40.4^{\circ})}) = 349\).

Step3: Calculate the values of the tangent - related terms

We know that \(\tan(25.7^{\circ})\approx0.483\) and \(\tan(40.4^{\circ})\approx0.859\). Then \(\frac{1}{\tan(25.7^{\circ})}\approx\frac{1}{0.483}\approx2.07\) and \(\frac{1}{\tan(40.4^{\circ})}\approx\frac{1}{0.859}\approx1.16\). So \(\frac{1}{\tan(25.7^{\circ})}+\frac{1}{\tan(40.4^{\circ})}\approx2.07 + 1.16=3.23\).

Step4: Solve for \(h\)

\(h=\frac{349}{3.23}\approx108\).

Answer:

108