Heights and Distances
From a point on the ground, the angle of elevation of the top of a tower is 45°. If the point is 30 m from the base of the tower, find the height of the tower.
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Step-by-Step Solution
Step 1: Identify the given information and unknown
We are given the angle of elevation from a point on the ground to the top of a tower, which is 45∘. We also know the distance from this point to the base of the tower, which is 30 m. We need to find the height of the tower. Let's represent the height of the tower as h.
Step 2: Formulate the trigonometric relationship
In the right-angled triangle formed by the tower, the ground, and the line of sight, the height of the tower (AB) is the side opposite to the angle of elevation, and the distance from the base (BC) is the adjacent side. The tangent function relates the opposite side to the adjacent side.
Step 3: Substitute values and solve for height
We know that tan45∘=1. Substituting this value and the given distance BC=30 m into our equation, we can solve for h. Multiplying both sides by 30 gives us the height of the tower.