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From a point on a bridge across a river,...

From a point on a bridge across a river, the angles of depression of the banks on opposite sides, of the river are 30° and 45° respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.

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To solve the problem, we will use the concept of angles of depression and right-angled triangles. Let's break it down step by step. ### Step 1: Understand the Problem We have a bridge at a height of 3 meters above the banks of a river. The angles of depression to the banks on either side of the river are 30° and 45°. We need to find the width of the river. ### Step 2: Draw the Diagram 1. Draw a horizontal line representing the bridge. 2. Mark the height of the bridge as 3 meters above the river banks. 3. From the point on the bridge, draw lines down to the banks at angles of 30° and 45°. ### Step 3: Identify the Right Triangles From the point on the bridge (let's call it point A), drop perpendiculars to the banks at points B and C. - Triangle ADB (where D is the point directly below A on the bank opposite to B) will have an angle of depression of 30°. - Triangle AEC (where E is the point directly below A on the bank opposite to C) will have an angle of depression of 45°. ### Step 4: Use Trigonometry to Find Distances 1. **For Triangle ADB (30° angle):** - The height (AD) = 3 m. - Using the tangent function: \[ \tan(30°) = \frac{AD}{DB} \] - Therefore, \[ \tan(30°) = \frac{3}{DB} \implies DB = \frac{3}{\tan(30°)} = 3 \cdot \sqrt{3} \] 2. **For Triangle AEC (45° angle):** - The height (AD) = 3 m. - Using the tangent function: \[ \tan(45°) = \frac{AD}{EC} \] - Therefore, \[ \tan(45°) = \frac{3}{EC} \implies EC = \frac{3}{\tan(45°)} = 3 \] ### Step 5: Calculate the Width of the River The width of the river (BC) is the sum of distances DB and EC: \[ BC = DB + EC = (3 \sqrt{3}) + 3 \] ### Step 6: Final Calculation To find the numerical value: \[ BC = 3\sqrt{3} + 3 \approx 3(1.732) + 3 \approx 5.196 + 3 \approx 8.196 \text{ meters} \] ### Conclusion The width of the river is approximately 8.2 meters. ---
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