A 1.5 m tall boy is standing at some distance from a 30 m 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. (trigonometry, angle of elevation, right triangle, distance)
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Step-by-Step Solution
Step 1: Determine the effective height of the building
First, we need to find the effective height of the building from the boy's eye level. Since the boy is 1.5 m tall and the building is 30 m tall, the height we consider for the angle of elevation calculations is the difference between these two values.
Step 2: Calculate the initial distance from the building
Let x1 be the initial distance of the boy from the building. Using the tangent function in the right-angled triangle formed by the boy's initial position, his eye level, and the top of the building, we can write tan(30∘)=adjacentopposite=x128.5. We know that tan(30∘)=31.
Step 3: Calculate the final distance from the building
Let x2 be the final distance of the boy from the building after walking towards it. Similarly, using the tangent function for the new angle of elevation, we have tan(60∘)=x228.5. We know that tan(60∘)=3.
Step 4: Calculate the distance walked
The distance the boy walked towards the building is the difference between his initial distance (x1) and his final distance (x2) from the building. We substitute the values we found for x1 and x2 and simplify the expression.