A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be $60^\circ$. Find the height of the tower.

Step 1: Draw a diagram and identify components

To solve this problem, we first draw a simple right-angled triangle representing the situation. Let AB represent the tower, and let C be the point on the ground from which the angle of elevation is measured. The distance from the foot of the tower to point C is BC.

Step 2: Identify given values

We are given that the angle of elevation to the top of the tower from point C is $60^\circ$, which means $\angle ACB = 60^\circ$. The distance from the foot of the tower (B) to the point C is given as 15 m.

Step 3: Apply trigonometric ratio

In the right-angled triangle ABC, we need to find the height of the tower, AB. We know the angle of elevation and the adjacent side (BC). The trigonometric ratio that relates the opposite side (AB) to the adjacent side (BC) is the tangent function.

Step 4: Substitute values and solve for height

Substitute the known values into the tangent equation. The value of $\tan 60^\circ$ is $\sqrt{3}$. We can then solve for AB, which represents the height of the tower. This gives us the exact height of the tower.

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