Question

question
a boat heading out to sea starts out at point a, at a horizontal distance of 1465 feet from a lighthouse/the shore. from that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon - light from that point to be 13°. 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 foot if necessary.

Explanation:

Step1: Find the height of the lighthouse

Let the height of the lighthouse be \(h\).
Using the tangent function at point \(A\): \(\tan13^{\circ}=\frac{h}{1465}\)
So, \(h = 1465\times\tan13^{\circ}\)
\(h=1465\times0.230868\approx338.22\)

Step2: Find the horizontal distance from point \(B\) to the lighthouse

Let the horizontal distance from point \(B\) to the lighthouse be \(x\).
Using the tangent function at point \(B\): \(\tan8^{\circ}=\frac{h}{x}\)
Since \(h\approx338.22\), then \(x=\frac{338.22}{\tan8^{\circ}}\)
\(\tan8^{\circ}\approx0.140541\), so \(x=\frac{338.22}{0.140541}\approx2406.5\)

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

The distance from \(A\) to \(B\) is \(x - 1465\)
\(2406.5-1465 = 941.5\approx942\)

Answer:

\(942\)