A yacht is anchored 90 feet offshore from the base of a lighthouse. The angle of elevation from the boat to the top of the lighthouse is 26 degrees. The distance between the yacht and the top of the lighthouse is about 100 feet. Calculate the height of the lighthouse? (Round your answer to the nearest whole number.)
Question
A yacht is anchored 90 feet offshore from the base of a lighthouse. The angle of elevation from the boat to the top of the lighthouse is 26 degrees. The distance between the yacht and the top of the lighthouse is about 100 feet. Calculate the height of the lighthouse? (Round your answer to the nearest whole number.)
Solution
To solve this problem, we can use trigonometry. Specifically, we'll use the tangent of the angle of elevation, which is the ratio of the opposite side (the height of the lighthouse) to the adjacent side (the distance from the yacht to the base of the lighthouse).
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We know that the angle of elevation is 26 degrees and the distance from the yacht to the base of the lighthouse is 90 feet.
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The formula for the tangent of an angle is: tan(angle) = opposite/adjacent.
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We can rearrange this formula to solve for the height of the lighthouse (the opposite side): height = tan(angle) * distance.
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Plugging in the values we know: height = tan(26 degrees) * 90 feet.
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Using a calculator, tan(26 degrees) is approximately 0.4877.
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So, the height of the lighthouse is approximately 0.4877 * 90 feet = 43.89 feet.
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Rounding to the nearest whole number, the height of the lighthouse is approximately 44 feet.
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