A 1.5m tall boy is standing at some distance from a 32m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.
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Step-by-Step Solution
Step 1: Identify knowns and set up the diagram
Let PQ be the height of the building and RS be the height of the boy. The boy's eye level is RS=1.5 m from the ground. We need to find the height of the building from the boy's eye level, which is PT. We subtract the boy's height from the building's height to get PT=32−1.5=30.5 m.
Step 2: Use trigonometry in the first triangle
Consider the right-angled triangle △PTU, where PT is the height of the building from the boy's eye level, and TU is the initial distance from the boy to the building. The angle of elevation is 30∘. We use the tangent function, tanθ=adjacentopposite, to find TU.
Step 3: Use trigonometry in the second triangle
Next, consider the right-angled triangle △PTW, where PT is the same height, and TW is the distance from the boy to the building after he walks closer. The new angle of elevation is 60∘. Again, we use the tangent function to find TW.
Step 4: Calculate the distance walked
The distance the boy walked towards the building is the difference between his initial distance from the building (TU) and his final distance from the building (TW). We subtract TW from TU to find UW.