A boat is heading towards a lighthouse, whose beacon-light is 111 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 11degrees
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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 21degrees
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. Find the distance from point AA to point BB. Round your answer to the nearest tenth of a foot if necessary.
Let's assume that the distance from point AA to the lighthouse is x feet.
From point AA, the boat's crew measures the angle of elevation to the beacon as 11 degrees. This means that the height of the lighthouse (111 feet) is the opposite side and the distance from AA to the lighthouse (x feet) is the adjacent side of the angle.
Using the tangent function, we can write the equation: tan(11 degrees) = opposite/adjacent
tan(11 degrees) = 111/x
To find the distance from point BB to the lighthouse, we can use the same approach. Let's assume the distance from point BB to the lighthouse is y feet.
From point BB, the boat's crew measures the angle of elevation to the beacon as 21 degrees. This means that the height of the lighthouse (111 feet) is the opposite side and the distance from BB to the lighthouse (y feet) is the adjacent side of the angle.
Using the tangent function, we can write the equation: tan(21 degrees) = opposite/adjacent
tan(21 degrees) = 111/y
We need to find the distance from point AA to point BB, which is the sum of x and y.
To find x and y, we can solve the two equations we derived:
x = 111/tan(11 degrees)
y = 111/tan(21 degrees)
Substituting the given values and calculating:
x ≈ 111/tan(11 degrees) ≈ 573.8 feet
y ≈ 111/tan(21 degrees) ≈ 293.9 feet
Finally, the distance from point AA to point BB is: x + y ≈ 573.8 + 293.9 ≈ 867.7 feet
Therefore, the distance from point AA to point BB is approximately 867.7 feet.