Question

a boat heading out to sea starts out at point a, at a horizontal distance of 1149 feet from a lighthouse/the shore. from that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon - light to be 16°. at some later time, the crew measures the angle of elevation from point b to be 8°. find the distance from point a to point b. round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Find the height of the lighthouse (h)

Let \( h \) be the height of the lighthouse. From point \( A \), the angle of elevation is \( 16^\circ \) and the horizontal distance from \( A \) to the lighthouse (\( L \)) is 1149 feet. Using the tangent function: \( \tan(16^\circ)=\frac{h}{1149} \), so \( h = 1149\times\tan(16^\circ) \).
\( h\approx1149\times0.2867\approx329.4 \) feet.

Step2: Find the horizontal distance from \( B \) to \( L \) (let's call it \( x \))

From point \( B \), the angle of elevation is \( 8^\circ \). Using the tangent function again: \( \tan(8^\circ)=\frac{h}{x} \), so \( x=\frac{h}{\tan(8^\circ)} \).
Substituting \( h\approx329.4 \): \( x\approx\frac{329.4}{\tan(8^\circ)}\approx\frac{329.4}{0.1405}\approx2344.5 \) feet.

Step3: Find the distance from \( A \) to \( B \)

The distance from \( A \) to \( B \) is \( x - 1149 \). So \( AB=2344.5 - 1149 = 1195.5 \) feet.

Answer:

1195.5