To solve the problem, we need to calculate the horizontal distance from the airplane to the air traffic control tower. Here's the setup:

Known values:

1. Altitude of the airplane = 7450 ft.
2. Height of the tower = 320 ft.
3. Angle of depression = $56^\circ$.

The horizontal distance can be found using the following steps:

Step 1: Find the vertical distance between the airplane and the top of the control tower.

$\text{Vertical distance} = \text{Altitude of airplane} - \text{Height of tower}$ $\text{Vertical distance} = 7450 - 320 = 7130 \, \text{ft.}$

Step 2: Use the tangent function to find the horizontal distance.

The tangent of the angle of depression relates the vertical distance and the horizontal distance: $\tan(\theta) = \frac{\text{Vertical distance}}{\text{Horizontal distance}}$ Rearranging: $\text{Horizontal distance} = \frac{\text{Vertical distance}}{\tan(\theta)}$

Substitute the values: $\text{Horizontal distance} = \frac{7130}{\tan(56^\circ)}$

Step 3: Compute the result.

Using a calculator: $\tan(56^\circ) \approx 1.4826$ $\text{Horizontal distance} = \frac{7130}{1.4826} \approx 4810.5 \, \text{ft.}$

Final Answer:

The horizontal distance from the airplane to the control tower is approximately 4810.5 feet.

Would you like me to clarify any of these steps?

Related Questions:

1. What is the role of the angle of depression in solving this problem?
2. How would the calculation change if the airplane's altitude were increased?
3. What is the trigonometric relationship between horizontal and vertical distances in a right triangle?
4. How could you visualize this problem using a diagram?
5. What if the height of the tower were unknown? How could you solve for it?

Tip:

Always ensure your calculator is in the correct mode (degrees or radians) based on the angle given in the problem.