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The image you provided shows a right triangle setup involving a radio tower, and the goal is to solve for the total height of the tower. The problem provides the following information:

- The length of the horizontal side adjacent to the 25° angle is 87 feet.
- There is a right triangle where:
  - The angle between the ground and the diagonal cable is 40°.
  - The angle adjacent to the horizontal base is 25°.
  - The remaining angle at the top of the tower is 90° (since it is a right triangle).

We need to find the total height of the radio tower, which consists of two parts:
1. **The height of the lower part** (represented by the shorter side of the triangle).
2. **The height of the upper part (x)**, which is given by the cable reaching upward from the top of the lower part at an angle of 40°.

### Step 1: Find the lower height using trigonometry
In the triangle, the 25° angle is adjacent to the known length of 87 feet, and we are interested in the opposite side (the lower part of the tower's height).

We can use the tangent function:
\[
\tan(25^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h_{\text{lower}}}{87}
\]
Solving for \( h_{\text{lower}} \):
\[
h_{\text{lower}} = 87 \cdot \tan(25^\circ)
\]
\[
h_{\text{lower}} \approx 87 \cdot 0.4663 \approx 40.57 \text{ feet}
\]

### Step 2: Use the 40° angle to find the upper height (x)
The second part involves solving for the height \( x \), which is the vertical side opposite the 40° angle. We use the sine function here:
\[
\sin(40^\circ) = \frac{x}{\text{hypotenuse}} = \frac{x}{87}
\]
Solving for \( x \):
\[
x = 87 \cdot \sin(40^\circ)
\]
\[
x \approx 87 \cdot 0.6428 \approx 55.92 \text{ feet}
\]

### Step 3: Total height of the tower
Now that we have both the lower part and the upper part of the height, we can sum them to find the total height:
\[
\text{Total height} = h_{\text{lower}} + x \approx 40.57 + 55.92 \approx 96.49 \text{ feet}
\]

So, the total height of the radio tower is approximately **96.5 feet**.

Would you like further details on this process, or do you have any specific questions?

### Related Questions:
1. How does the tangent function relate to the sides of a right triangle?
2. Why was the sine function used in the second part of the problem?
3. Can you explain why the angles sum up to 90° in a right triangle setup?
4. What other trigonometric functions could be used to solve this problem?
5. How would the solution change if one of the angles was altered?

### Tip:
In right triangle problems, always identify which sides are opposite, adjacent, or hypotenuse relative to the angle you're using. This helps you choose the correct trigonometric function.

Solve for the TOTAL height of the radio tower using the given triangle with angles of 40° and 25° and a base of 87 feet.

Learn how to calculate the total height of a radio tower using trigonometric functions with given angles of 40° and 25°, and a base length of 87 feet. This problem involves applying sine and tangent formulas in a right triangle setup. Suitable for high school students in Grades 9-12.

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