A water tower is located x = 375 ft from a building (see the figure). From a window in the building, an observer notes that the angle of elevation to the top of the tower is 39° and that the angle of depression to the bottom of the tower is 25°

To solve this problem, we need to calculate the height of the water tower based on the information given:

- The distance from the building to the water tower is \( x = 375 \, \text{ft} \).
- The angle of elevation to the top of the tower is \( 39^\circ \).
- The angle of depression to the bottom of the tower is \( 25^\circ \).

Let's denote:
- \( h_{\text{top}} \) as the height of the water tower above the level of the observer.
- \( h_{\text{bottom}} \) as the height of the water tower below the observer's level.

### Step 1: Calculate \( h_{\text{top}} \)
Using the angle of elevation \( 39^\circ \), we can apply the tangent function:
\[
\tan(39^\circ) = \frac{h_{\text{top}}}{375}
\]
Solving for \( h_{\text{top}} \):
\[
h_{\text{top}} = 375 \times \tan(39^\circ)
\]

### Step 2: Calculate \( h_{\text{bottom}} \)
Using the angle of depression \( 25^\circ \), again using the tangent function:
\[
\tan(25^\circ) = \frac{h_{\text{bottom}}}{375}
\]
Solving for \( h_{\text{bottom}} \):
\[
h_{\text{bottom}} = 375 \times \tan(25^\circ)
\]

### Step 3: Total height of the tower
The total height of the water tower \( h \) is the sum of \( h_{\text{top}} \) and \( h_{\text{bottom}} \):
\[
h = h_{\text{top}} + h_{\text{bottom}}
\]

Let's calculate the values.The total height of the water tower is approximately \( 478.53 \, \text{ft} \).

Would you like more details or have any questions about this calculation?

Here are some related questions you might consider:
1. How can we determine the height of the observer's window in the building?
2. What other trigonometric functions can be applied in similar problems?
3. How would the answer change if the distance between the building and the tower increased?
4. How can we find the angle of elevation or depression if the tower height is given?
5. How does the observer’s position affect the angles used?

**Tip:** In trigonometry problems involving angles of elevation and depression, the tangent function is often useful when horizontal distances are known.

A water tower is located x = 375 ft from a building. From a window in the building, an observer notes that the angle of elevation to the top of the tower is 39° and that the angle of depression to the bottom of the tower is 25°.

Solve a real-world trigonometry problem involving a water tower 375 ft away from a building. Using angles of elevation (39°) and depression (25°), we calculate the total height of the tower using tangent functions. Ideal for students in Grades 10-12.

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