A surveyor is standing at point A and wants to measure the width of a river without crossing it. She measures the angles of elevation to two points on the opposite riverbank: 30 degrees and 45 degrees. If she is 100 meters from the closer bank and 150 meters from the farther bank, find the width of the river.
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Let's call the width of the river x.
From point A, the surveyor measures the angle of elevation to the closer bank as 30 degrees.
Using trigonometry, we can set up the following equation: tan(30) = x / 100
tan(30) = √3 / 3, so we can rewrite the equation as:
√3 / 3 = x / 100
Multiplying both sides by 100: x = 100 * (√3 / 3)
x = 100√3 / 3
From point A, the surveyor measures the angle of elevation to the farther bank as 45 degrees.
Using trigonometry again, we can set up the following equation: tan(45) = x / 150
tan(45) = 1, so we can rewrite the equation as:
1 = x / 150
Multiplying both sides by 150: x = 150
Therefore, the width of the river is x = 100√3 / 3 = 150 meters.
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