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).

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