Question

1. zack is standing at the top of a lookout tower and spots a water fountain below. if the lookout tower is 75 feet tall and the angle of depression is 28°, what is the horizontal distance between zack and the water fountain?

Explanation:

Step1: Analyze the angle relationship

The angle of depression is \(28^{\circ}\), and the angle of depression is equal to the angle of elevation from the fountain to the top of the tower. Let the horizontal distance be \(x\). We know the height of the tower \(h = 75\) feet.

Step2: Use the tangent function

We use the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 28^{\circ}\), the opposite side is the height of the tower (\(h = 75\) feet), and the adjacent side is the horizontal distance \(x\). So \(\tan28^{\circ}=\frac{75}{x}\).

Step3: Solve for \(x\)

We can rewrite the equation as \(x=\frac{75}{\tan28^{\circ}}\). Since \(\tan28^{\circ}\approx0.5317\), then \(x=\frac{75}{0.5317}\approx141.5\) feet.

Answer:

$141.5$ feet