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.

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 $45^\circ$. We also know the distance from this point to the base of the tower, which is $30 \text{ m}$. 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 $\tan 45^\circ = 1$. Substituting this value and the given distance $BC = 30 \text{ m}$ into our equation, we can solve for $h$. Multiplying both sides by $30$ gives us the height of the tower.

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