An airplane is at a horizontal distance of 1050 m from a control tower. The angles of depression from the airplane to the top and base of the tower are $36^\circ$ and $41^\circ$ respectively. We need to calculate the height of the control tower and the shortest distance between the airplane and the base of the control tower.

1. Problem Description

An airplane is at a horizontal distance of 1050 m from a control tower. The angles of depression from the airplane to the top and base of the tower are $36^\circ$ and $41^\circ$ respectively. We need to calculate the height of the control tower and the shortest distance between the airplane and the base of the control tower.

2. Solution Steps

Let $h$ be the height of the control tower, and $x$ be the vertical distance from the airplane to the top of the tower. Let $d$ be the horizontal distance from the airplane to the tower, which is given as 1050 m.

Let $\theta_1$ be the angle of depression to the top of the tower, and $\theta_2$ be the angle of depression to the base of the tower. We are given that $\theta_1 = 36^\circ$ and $\theta_2 = 41^\circ$.

The vertical distance from the airplane to the base of the tower is $x+h$.

We can use the tangent function to relate the angles of depression to the distances:

$tan(\theta_1) = \frac{x}{d}$, so $tan(36^\circ) = \frac{x}{1050}$.

$tan(\theta_2) = \frac{x+h}{d}$, so $tan(41^\circ) = \frac{x+h}{1050}$.

First, we find $x$:

$x = 1050 \cdot tan(36^\circ) = 1050 \cdot 0.7265 \approx 762.825$

Next, we find $x+h$:

$x+h = 1050 \cdot tan(41^\circ) = 1050 \cdot 0.8693 \approx 912.765$

Now, we can find $h$:

$h = (x+h) - x = 912.765 - 762.825 = 149.94$

So, the height of the tower is approximately 150 m.

For the shortest distance between the airplane and the base of the control tower, we can use the Pythagorean theorem or the cosine function. Let $D$ be this distance. We have:

$cos(\theta_2) = \frac{d}{D}$

$D = \frac{d}{cos(\theta_2)} = \frac{1050}{cos(41^\circ)} = \frac{1050}{0.7547} \approx 1391.28$

Alternatively, we can use the Pythagorean theorem:

$D = \sqrt{d^2 + (x+h)^2} = \sqrt{1050^2 + 912.765^2} = \sqrt{1102500 + 833180.5} = \sqrt{1935680.5} \approx 1391.29$

So, the shortest distance between the airplane and the base of the control tower is approximately 1391 m.

3. Final Answer

(i) The height of the control tower is 150 m.

(ii) The shortest distance between the airplane and the base of the control tower is 1391 m.