Question

a boat heading out to sea starts out at point a, at a horizontal distance of 1390 feet from a lighthouse/the shore. from that point, the boats crew measures the angle of elevation to the lighthouses beacon - light to be 15°. at some later time, the crew measures the angle of elevation from point b to be 7°. 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 \(h\). In right - triangle \(AL\) (where \(AL = 1390\) feet and the angle of elevation from \(A\) is \(15^{\circ}\)), we use the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
So, \(\tan15^{\circ}=\frac{h}{1390}\), then \(h = 1390\times\tan15^{\circ}\).
We know that \(\tan15^{\circ}=2 - \sqrt{3}\approx0.2679\), so \(h=1390\times0.2679 = 372.381\) feet.

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

Let the distance from \(B\) to the lighthouse be \(x\). In right - triangle \(BL\) (where the angle of elevation from \(B\) is \(7^{\circ}\) and \(h\) is the height of the lighthouse), using the tangent function \(\tan7^{\circ}=\frac{h}{x}\).
Since \(h = 372.381\) feet and \(\tan7^{\circ}\approx0.1228\), then \(x=\frac{h}{\tan7^{\circ}}=\frac{372.381}{0.1228}\approx3032.42\) feet.

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

The distance from \(A\) to \(B\) is \(x - 1390\).
Substitute \(x\approx3032.42\) into \(x - 1390\), we get \(3032.42-1390=1642.42\approx1642\) feet.

Answer:

\(1642\) feet