Question

a boat is heading towards a lighthouse, whose beacon - light is 107 feet above the water. from point a, the boat’s crew measures the angle of elevation to the beacon, 5°, before they draw closer. they measure the angle of elevation a second time from point b to be 11°. find the distance from point a to point b. round your answer to the nearest foot if necessary.

Explanation:

Step1: Find distance from A to L

We know that in right triangle \( \triangle AL L_{beacon} \) (where \( L_{beacon} \) is the top of the lighthouse), \( \tan(5^{\circ})=\frac{107}{AL} \). So, \( AL = \frac{107}{\tan(5^{\circ})} \). Using a calculator, \( \tan(5^{\circ})\approx0.0875 \), so \( AL\approx\frac{107}{0.0875}\approx1222.86 \) feet.

Step2: Find distance from B to L

In right triangle \( \triangle B L L_{beacon} \), \( \tan(11^{\circ})=\frac{107}{BL} \). So, \( BL=\frac{107}{\tan(11^{\circ})} \). Using a calculator, \( \tan(11^{\circ})\approx0.1944 \), so \( BL\approx\frac{107}{0.1944}\approx550.41 \) feet.

Step3: Find distance from A to B

The distance \( AB = AL - BL \). Substituting the values, \( AB\approx1222.86 - 550.41 = 672.45\approx672 \) feet.

Answer:

\( 672 \)