Question

a boat heading out to sea starts out at point a, at a horizontal distance of 724 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 16°. at some later time, the crew measures the angle of elevation from point b to be 5°. 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 (h)

Let the height of the lighthouse be \( h \). At point \( A \), the horizontal distance from the lighthouse is \( 724 \) feet and the angle of elevation is \( 16^\circ \). Using the tangent function (\( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \)):
\( \tan(16^\circ)=\frac{h}{724} \)
So, \( h = 724\times\tan(16^\circ) \). Calculating \( \tan(16^\circ)\approx0.2867 \), then \( h\approx724\times0.2867\approx207.67 \) feet.

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

At point \( B \), the angle of elevation is \( 5^\circ \) and the height of the lighthouse is still \( h \). Using the tangent function again:
\( \tan(5^\circ)=\frac{h}{x} \)
We know \( h\approx207.67 \), so \( x=\frac{h}{\tan(5^\circ)} \). Calculating \( \tan(5^\circ)\approx0.0875 \), then \( x\approx\frac{207.67}{0.0875}\approx2373.37 \) feet.

Step3: Find the distance from A to B

The distance from \( A \) to \( B \) is the difference between the horizontal distance from \( B \) to the lighthouse and from \( A \) to the lighthouse. Let \( d \) be the distance from \( A \) to \( B \):
\( d=x - 724 \)
Substituting \( x\approx2373.37 \) and \( 724 \):
\( d\approx2373.37 - 724 = 1649.37 \approx 1649 \) feet.

Answer:

The distance from point \( A \) to point \( B \) is approximately \(\boxed{1649}\) feet.