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=(AdjacentLeg)/(OppositeLeg).`

Therefore, using the above diagram we get:

`\begin{gathered} \cot7º=(AC)/(104ft), \\ \cot24^(\circ)=(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.