A water tower is located 441 feet from a building. From a window in the building, an observer notes that the angle of elevation to the top of the tower is $$ 37^{\circ} $$ and that the angle of depression to the bottom of the tower is $$ 26^{\circ} $$. How tall is the tower? height $$ = $$ How high is the window from the ground? window is Report answer accurate to 2 decimal places. If it helps, there is a similar picture on page 486 of your Stewart text (see \#55).

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To determine the height of the water tower and the height of the window from the ground, we can use trigonometric relationships involving the angles of elevation and depression.

**Given:**
- Horizontal distance from the building to the tower ($$d$$) = 441 feet
- Angle of elevation to the top of the tower ($$\theta$$) = $$37^\circ$$
- Angle of depression to the bottom of the tower ($$\phi$$) = $$26^\circ$$

**Let:**
- $$h$$ = Height of the tower
- $$y$$ = Height of the window from the ground

**Step 1: Calculate the height of the window ($$y$$).**

Using the angle of depression:
$$
\tan(\phi) = \frac{y}{d}
$$
$$
y = d \cdot \tan(\phi) = 441 \cdot \tan(26^\circ) \approx 441 \cdot 0.487732 \approx 215.09 \text{ feet}
$$

**Step 2: Calculate the total height of the tower ($$h$$).**

Using the angle of elevation:
$$
\tan(\theta) = \frac{h - y}{d}
$$
$$
h - y = d \cdot \tan(\theta) = 441 \cdot \tan(37^\circ) \approx 441 \cdot 0.753554 \approx 332.31 \text{ feet}
$$
$$
h = 332.31 + 215.09 \approx 547.40 \text{ feet}
$$

**Final Answers:**
- **Height of the tower:** $$547.41$$ feet
- **Height of the window from the ground:** $$215.09$$ feet
The tower is approximately 547.41 feet tall, and the window is about 215.09 feet above the ground.

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