Question

A boat is heading towards a lighthouse, whose beacon-light is 104 feet above thewater. From point A, the boat's crew measures the angle of elevation to the beacon,7, before they draw closer. They measure the angle of elevation a second time frompoint B at some later time to be 24°. Find the distance from point A to point B.Round your answer to the nearest tenth of a foot if necessary.

Answer 1

To answer this question we will use the following diagram as a reference:

Recall that in a right triangle:

$\cot\theta=\frac{AdjacentLeg}{OppositeLeg}.$

Therefore, using the above diagram we get:

$\begin{gathered} \cot7º=\frac{AC}{104ft}, \\ \cot24^{\circ}=\frac{BC}{104ft}. \end{gathered}$

Then:

$\begin{gathered} AC=104\cot7^{\circ}ft, \\ BC=104\cot24^{\circ}ft. \end{gathered}$

Now, notice that:

$AB=AC-BC.$

Then:

$AB=104\cot7^{\circ}ft-104\cot24^{\circ}ft.$

Simplifying the above result we get:

$AB\approx847.0ft-233.6ft=613.4ft$

Answer: 613.4 ft.