From a point on a bridge across a river the angle of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at the height of 30 m from the banks, the width of the river is
Question
From a point on a bridge across a river the angle of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at the height of 30 m from the banks, the width of the river is
Solution
To solve this problem, we can use the concept of trigonometry.
Step 1: Identify the given information. The height of the bridge is 30m, the angle of depression to one bank is 30°, and the angle of depression to the other bank is 45°.
Step 2: Use the tangent of the angles of depression. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
Step 3: Calculate the distance from the point on the bridge to each bank.
For the bank with the 30° angle of depression, tan(30°) = height / distance. So, distance = height / tan(30°) = 30m / tan(30°) = 30m / √3/3 = 30√3 m.
For the bank with the 45° angle of depression, tan(45°) = height / distance. So, distance = height / tan(45°) = 30m / 1 = 30m.
Step 4: The width of the river is the sum of these two distances. So, the width of the river = 30m + 30√3 m.
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