7. At ground level, a water tower is 430 feet from the base of a building. From one of the upper floors of the building the angle of elevation to the top of the water tower is 15 and the angle of depression to the bottom of the water tower is 28 . How tall is the water tower? How hgh off the ground is the observer?

To find the height of the water tower and the observer's height, we can use some trigonometry. Let's denote the height of the observer from the ground as \( h \) and the total height of the water tower as \( H \). The observer is at a height \( h \), looking at the top of the water tower at an angle of elevation of 15 degrees. The horizontal distance to the tower is 430 feet. Using the tangent function: \[ \tan(15^\circ) = \frac{H - h}{430} \] From this, we can express \( H \) in terms of \( h \): \[ H - h = 430 \tan(15^\circ) \] \[ H = h + 430 \tan(15^\circ) \] Now for the bottom of the water tower, with an angle of depression of 28 degrees: \[ \tan(28^\circ) = \frac{h}{430} \] From this, we can solve for \( h \): \[ h = 430 \tan(28^\circ) \] Now, we can calculate \( H \): 1. Calculate \( h \): \[ h = 430 \tan(28^\circ) \approx 430 \times 0.5317 \approx 228.63 \text{ feet} \] 2. Plug \( h \) into the equation for \( H \): \[ H = 228.63 + 430 \tan(15^\circ) \approx 228.63 + 430 \times 0.2679 \approx 228.63 + 115.59 \approx 344.22 \text{ feet} \] So, the water tower is approximately 344.22 feet tall, and the observer is about 228.63 feet off the ground.