Question

a boat heading out to sea starts out at point a, at a horizontal distance of 1023 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 11°. at some later time, the crew measures the angle of elevation from point b to be 2°. 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 \( h \). From point \( A \), the horizontal distance to the lighthouse is 1023 feet and the angle of elevation is \( 11^\circ \). Using the tangent function: \( \tan(11^\circ)=\frac{h}{1023} \), so \( h = 1023\times\tan(11^\circ) \).

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

From point \( B \), the angle of elevation is \( 2^\circ \). Using the tangent function again: \( \tan(2^\circ)=\frac{h}{x} \), so \( x=\frac{h}{\tan(2^\circ)} \). Substitute \( h \) from Step 1: \( x = \frac{1023\times\tan(11^\circ)}{\tan(2^\circ)} \).

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

The distance from \( A \) to \( B \) is \( x - 1023 \). First, calculate \( h = 1023\times\tan(11^\circ)\approx1023\times0.1944\approx200.9 \) feet. Then, \( x=\frac{200.9}{\tan(2^\circ)}\approx\frac{200.9}{0.0349}\approx5756 \) feet. Finally, the distance \( AB = 5756 - 1023 = 4733 \) feet.

Answer:

4733