The problem describes a scenario where the angle of depression from the top of a building to a point $P$ on the ground is $23.6^\circ$. The distance from point $P$ to the foot of the building is 50 meters. We need to find the height of the building to the nearest meter.

1. Problem Description

The problem describes a scenario where the angle of depression from the top of a building to a point $P$ on the ground is $23.6^\circ$. The distance from point $P$ to the foot of the building is 50 meters. We need to find the height of the building to the nearest meter.

2. Solution Steps

Let $h$ be the height of the building. Let $d$ be the distance from point $P$ to the foot of the building, which is given as $d=50$ meters. The angle of depression from the top of the building to point $P$ is $23.6^\circ$. The angle of elevation from point $P$ to the top of the building is equal to the angle of depression.

We can use the tangent function to relate the height of the building to the distance $d$ and the angle of elevation.

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

In this case, the angle of elevation is $\theta = 23.6^\circ$, the opposite side is the height of the building $h$, and the adjacent side is the distance $d = 50$ meters.

So, we have:

$$\tan(23.6^\circ) = \frac{h}{50}$$

We can solve for $h$ by multiplying both sides by 50:

$$h = 50 \cdot \tan(23.6^\circ)$$

Using a calculator, we find that $\tan(23.6^\circ) \approx 0.43696$.

$$h \approx 50 \cdot 0.43696 \approx 21.848$$

Rounding to the nearest meter, we get $h \approx 22$ meters.

3. Final Answer

The height of the building is approximately 22 meters.