Question

a boat heading out to sea starts out at point a, at a horizontal distance of 1290 feet from a lighthouse/the shore. from that point, the boats crew measures the angle of elevation to the lighthouses beacon - light from that point to be 6°. at some later time, the crew measures the angle of elevation from point b to be 3°. find the distance from point a to point b. round your answer to the nearest foot if necessary.

Explanation:

Step1: Set up equations using tangent function

Let the height of the lighthouse be \(h\).
From point \(A\) (distance \(d_A = 1290\) feet), \(\tan6^{\circ}=\frac{h}{1290}\), so \(h = 1290\times\tan6^{\circ}\).
From point \(B\) (distance \(d_B\)), \(\tan3^{\circ}=\frac{h}{d_B}\), so \(h = d_B\times\tan3^{\circ}\).
Since \(h\) is the same in both cases, \(1290\times\tan6^{\circ}=d_B\times\tan3^{\circ}\).

Step2: Solve for \(d_B\)

We know that \(\tan6^{\circ}\approx0.1051\) and \(\tan3^{\circ}\approx0.0524\).
Substitute the values into the equation \(1290\times0.1051=d_B\times0.0524\).
First, calculate \(1290\times0.1051 = 135.579\).
Then, \(d_B=\frac{135.579}{0.0524}\).
\(d_B\approx2587.4\).
The distance from \(A\) to \(B\) is \(d = d_B - 1290\).
\(d=2587.4 - 1290=1297.4\approx1297\)

Answer:

1297