Question

a lighthouse sits at the edge of a cliff, as shown. a ship at sea level is 550 meters from the base of the cliff. the angle of elevation from sea level to the base of the lighthouse is 62.3°, the angle of elevation from sea level to the top of the lighthouse is 65.1°. find the height of the lighthouse from the top of the cliff. do not round any intermediate computations. round your answer to the nearest tenth. note that the figure below is not drawn to scale.

Explanation:

Step1: Calculate the height from sea - level to the top of the lighthouse

We know that in a right - triangle, the tangent function is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
Let \(h_1\) be the height from sea - level to the top of the lighthouse. Given \(\theta = 65.1^{\circ}\) and the adjacent side \(x = 550\) meters.
Using the formula \(\tan\theta=\frac{h_1}{x}\), we have \(h_1=x\tan\theta\).
Substituting the values, \(h_1 = 550\times\tan(65.1^{\circ})\).
Since \(\tan(65.1^{\circ})\approx2.15\), then \(h_1=550\times2.15 = 1182.5\) meters.

Step2: Calculate the height from sea - level to the base of the lighthouse

Let \(h_2\) be the height from sea - level to the base of the lighthouse. Given \(\theta = 62.3^{\circ}\) and the adjacent side \(x = 550\) meters.
Using the formula \(\tan\theta=\frac{h_2}{x}\), we have \(h_2=x\tan\theta\).
Since \(\tan(62.3^{\circ})\approx1.9\), then \(h_2 = 550\times1.9=1045\) meters.

Step3: Calculate the height of the lighthouse

The height of the lighthouse \(h=h_1 - h_2\).
\(h=1182.5-1045\)
\(h = 137.5\) meters.

Answer:

\(137.5\) meters