Question

find h to the nearest tenth. the length of h is approximately
(round the final answer to the nearest tenth as needed. round all intermediate values to the nearest tenth as needed.)

Explanation:

Step1: Express \( AD \) in terms of \( h \)

In right - triangle \( ABD \), \( \tan25^{\circ}=\frac{h}{AD}\), so \( AD = \frac{h}{\tan25^{\circ}}\). In right - triangle \( BCD \), \( \tan48^{\circ}=\frac{h}{BD}\), so \( BD=\frac{h}{\tan48^{\circ}}\). And we know that \( AD - BD=275\).

Step2: Substitute \( AD \) and \( BD \) into the equation

Substitute \( AD=\frac{h}{\tan25^{\circ}}\) and \( BD = \frac{h}{\tan48^{\circ}}\) into \( AD - BD = 275\), we get \( h(\frac{1}{\tan25^{\circ}}-\frac{1}{\tan48^{\circ}})=275\).
Since \(\tan25^{\circ}\approx0.4663\), \(\tan48^{\circ}\approx1.1106\), then \(\frac{1}{\tan25^{\circ}}-\frac{1}{\tan48^{\circ}}=\frac{\tan48^{\circ}-\tan25^{\circ}}{\tan25^{\circ}\tan48^{\circ}}\approx\frac{1.1106 - 0.4663}{0.4663\times1.1106}\approx\frac{0.6443}{0.5179}\approx1.244\).

Step3: Solve for \( h \)

From \( h\times1.244 = 275\), we can solve for \( h\) as \( h=\frac{275}{1.244}\approx221.1\).

Answer:

\(221.1\)