Question

a boat heading out to sea starts out at point a, at a horizontal distance of 1046 feet from a lighthouse/the shore. from that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon - light to be 15°. at some later time, the crew measures the angle of elevation from point b to be 6°. 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)

We know that at point A, the horizontal distance from the lighthouse (L) is \( AL = 1046 \) feet and the angle of elevation is \( 15^\circ \). Using the tangent function, \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \), so \( \tan(15^\circ)=\frac{h}{1046} \). Then \( h = 1046\times\tan(15^\circ) \). Calculating \( \tan(15^\circ)\approx0.2679 \), so \( h\approx1046\times0.2679\approx280.3 \) feet.

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

At point B, the angle of elevation is \( 6^\circ \) and the height of the lighthouse is \( h\approx280.3 \) feet. Using the tangent function again, \( \tan(6^\circ)=\frac{h}{BL} \), so \( BL=\frac{h}{\tan(6^\circ)} \). Calculating \( \tan(6^\circ)\approx0.1051 \), then \( BL\approx\frac{280.3}{0.1051}\approx2667.0 \) feet.

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

We know that \( BL = AB + AL \), so \( AB = BL - AL \). Substituting the values, \( AB\approx2667.0 - 1046 = 1621.0 \) feet.

Answer:

1621