From a tower 150 feet high with the sun directly​ overhead, an airplane and its shadow have an angle of elevation of 66.5degrees and an angle of depression of 8.1degrees​, respectively. Find the height of the airplane.

We can solve this problem by breaking it down into two triangles and using trigonometric concepts.

### Given:
- The height of the tower \( T = 150 \) feet.
- The angle of elevation from the tower to the airplane \( \theta_1 = 66.5^\circ \).
- The angle of depression from the tower to the airplane's shadow \( \theta_2 = 8.1^\circ \).

We are tasked with finding the height of the airplane.

### Steps:

1. **Horizontal distance to the shadow**:
   First, let's find the horizontal distance from the base of the tower to the shadow of the airplane, which we will call \( d \).

   Since the angle of depression (downward) is measured from the top of the tower, we can use the tangent of \( \theta_2 \) for this right triangle:

   \[
   \tan(\theta_2) = \frac{T}{d}
   \]
   Solving for \( d \):
   \[
   d = \frac{T}{\tan(\theta_2)} = \frac{150}{\tan(8.1^\circ)}
   \]

2. **Height of the airplane**:
   Now, using the angle of elevation \( \theta_1 \), we can find the total height of the airplane, which we will call \( h_{\text{plane}} \). Again, using the tangent function:

   \[
   \tan(\theta_1) = \frac{h_{\text{plane}}}{d}
   \]
   Solving for \( h_{\text{plane}} \):
   \[
   h_{\text{plane}} = d \cdot \tan(\theta_1)
   \]

3. **Final substitution**:
   Now, substitute the value of \( d \) from step 1 into the equation for \( h_{\text{plane}} \).

Let's compute the values:

- \( \tan(8.1^\circ) \approx 0.1417 \), so:
  \[
  d = \frac{150}{0.1417} \approx 1058.4 \text{ feet}
  \]

- \( \tan(66.5^\circ) \approx 2.329 \), so:
  \[
  h_{\text{plane}} = 1058.4 \times 2.329 \approx 2464.5 \text{ feet}
  \]

Therefore, the height of the airplane is approximately **2464.5 feet** above the ground.

Would you like more details on any step or have questions?

Here are 5 related questions:
1. How do we calculate angles of depression and elevation in trigonometry?
2. Can you find the distance between the airplane and the tower?
3. How does the concept of tangent help in solving right triangles?
4. Can we solve for the length of the shadow cast by the airplane?
5. What is the difference between horizontal and vertical distance in such problems?

**Tip**: When dealing with angles of elevation or depression, always visualize a right triangle to identify which sides you need to calculate.

From a tower 150 feet high with the sun directly overhead, an airplane and its shadow have an angle of elevation of 66.5 degrees and an angle of depression of 8.1 degrees, respectively. Find the height of the airplane.

Solve a trigonometry problem involving a 150-foot-high tower, where an airplane's angle of elevation is 66.5 degrees and its shadow's angle of depression is 8.1 degrees. This solution demonstrates how to use tangent functions to find the height of the airplane, making it ideal for students in Grades 10-12.

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