A surveyor measures the distance across a river that flows straight north by the following method. Starting directly across from a tree on the opposite bank, the surveyor walks 100 m along the river to establish a baseline. She then sights across to the tree and reads that the angle from the baseline to the tree is 35°35° . How wide is the river?
Question
A surveyor measures the distance across a river that flows straight north by the following method. Starting directly across from a tree on the opposite bank, the surveyor walks 100 m along the river to establish a baseline. She then sights across to the tree and reads that the angle from the baseline to the tree is 35°35° . How wide is the river?
Solution
To solve this problem, we can use trigonometry. Specifically, we can use the tangent of the angle, which is the ratio of the opposite side (the width of the river) to the adjacent side (the baseline).
Here are the steps:
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We know that the tangent of an angle in a right triangle is equal to the length of the opposite side divided by the length of the adjacent side. In this case, the angle is 35 degrees, the length of the adjacent side is 100 m (the baseline), and the length of the opposite side is the width of the river, which we're trying to find.
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So, we can set up the equation tan(35) = opposite/adjacent, or tan(35) = width of river/100.
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To solve for the width of the river, we multiply both sides of the equation by 100, giving us width of river = 100 * tan(35).
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Using a calculator, we find that tan(35) is approximately 0.7002.
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Therefore, the width of the river is approximately 100 * 0.7002 = 70.02 m.
So, the river is approximately 70.02 m wide.
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