Question

1. looking down from a 21 - foot tower, a forest ranger spots a fire on the ground. the angle of depression from the tower to the fire is 12°. a. draw a diagram to represent this situation. b. to the nearest foot, how far is the fire from the base of the tower? show all steps.

Explanation:

Step1: Analyze the right - triangle relationship

We know that the angle of depression is \(12^{\circ}\). The height of the tower (opposite side) \(h = 21\) feet. Let the distance from the base of the tower to the fire (adjacent side) be \(x\). Since the angle of depression is equal to the angle of elevation from the fire to the top of the tower. In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), where \(\theta = 12^{\circ}\) and the opposite side \(y = 21\) and adjacent side \(x\) (the distance we want to find).

Step2: Use the tangent formula

We have the formula \(\tan\theta=\frac{y}{x}\). Substituting \(\theta = 12^{\circ}\) and \(y = 21\), we get \(\tan(12^{\circ})=\frac{21}{x}\). Then \(x=\frac{21}{\tan(12^{\circ})}\).

We know that \(\tan(12^{\circ})\approx0.2126\). So \(x=\frac{21}{0.2126}\approx99\)

Answer:

The fire is approximately \(99\) feet from the base of the tower.