Question

A boat is heading towards a lighthouse, whose beacon-light is 108 feet above the water. From pointA, the boat’s crew measures the angle of elevation to the beacon, 8 degrees, before they draw closer. They measure the angle of elevation a second time from pointB at some later time to be 16∘. Find the distance from point A to point B. Round your answer to the nearest foot if necessary.

Answer 1

Given: The information of a boat heading towards a lighthouse

To Determine: The distance from point A to point B

Solution: The information provided can be translated into the diagram below

`\begin{gathered} m\angle ADC+m\angle DAC=90^0 \\ m\angle ADC+8^0=90^0 \\ m\angle ADC=90^0-8^0 \\ n\angle ADC=82^0 \end{gathered}`
`\begin{gathered} m\angle BDC+m\angle DBC=90^0 \\ m\angle BDC=90^0-m\angle DBC \\ m\angle BDC=90^0-16^0 \\ m\angle BDC=74^0 \end{gathered}`

Using SOH CAH TOA

`\begin{gathered} \tan 74^0=(BC)/(108) \\ BC=108\tan 74^0 \\ BC=108*3.4874 \\ BC=376.64ft \\ BC\approx377ft(nearest\text{ foot)} \end{gathered}`
`\begin{gathered} \tan 82^0=(AC)/(108) \\ AC=108*\tan 108 \\ AC=108*7.115 \\ AC=768.4599 \end{gathered}`
`\begin{gathered} AB=AC-BC \\ AB=768.45993-376.64 \\ AB=391.8199 \\ AB\approx392ft \end{gathered}`

Hence, the distance from point A to point B is 392ft (nearest foot)