Question

Boat is heading towards a lighthouse, whose beacon-light is 126 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 12^{\circ}

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, before they draw closer. They measure the angle of elevation a second time from point BB at some later time to be 25^{\circ}
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. Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

Answer 1

Using trigonometry with the given elevation angles and the height of the lighthouse's beacon-light, the distance between the two points AA and BB can be calculated by finding the lengths from each point to the lighthouse, and then subtracting them.

Step-by-step explanation:

The student's question involves finding the distance between point AA and point BB as a boat approaches a lighthouse. Given that the elevation angles change from 12 degrees to 25 degrees, and knowing the height of the beacon-light, we can use trigonometry to solve this problem. We will use the tangent function since it relates the angle of elevation to the opposite side (height of the beacon) and the adjacent side (distance from the point).

Let's call the distance from point AA to the lighthouse x and the distance from point BB to the lighthouse y. The height of the beacon-light is 126 feet.

For point AA, tan(12°) = 126 / x.
For point BB, tan(25°) = 126 / y.

Solving these equations gives us the values for x and y. The distance between point AA and BB is the difference between x and y, which can be simply calculated as x - y. After finding the numerical values, we round to the nearest foot as instructed.