Question

a lighthouse sits at the edge of a cliff, as shown. a ship at sea level is 1100 meters from the base of the cliff. the angle of elevation from sea level to the base of the lighthouse is 48.1°. the angle of elevation from sea level to the top of the lighthouse is 50.5°. 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. 1100 meters

Explanation:

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

We use the tangent function. For the angle of elevation of \(50.5^{\circ}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Let \(h_1\) be the height from sea - level to the top of the lighthouse.
\(\tan(50.5^{\circ})=\frac{h_1}{1100}\), so \(h_1 = 1100\times\tan(50.5^{\circ})\)
Using a calculator, \(\tan(50.5^{\circ})\approx1.20\), then \(h_1=1100\times1.20 = 1320\)

Step2: Find 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. For the angle of elevation of \(48.1^{\circ}\), \(\tan(48.1^{\circ})=\frac{h_2}{1100}\)
\(h_2 = 1100\times\tan(48.1^{\circ})\)
Using a calculator, \(\tan(48.1^{\circ})\approx1.11\), then \(h_2 = 1100\times1.11=1221\)

Step3: Find the height of the lighthouse from the top of the cliff

Let \(h\) be the height of the lighthouse from the top of the cliff. Then \(h=h_1 - h_2\)
\(h=1320 - 1221=99.0\)

Answer:

\(99.0\) meters