Question

from a fire tower 229 feet above ground level in the sasquatch national forest, a ranger spots a fire off in the distance. the angle of depression from the ranger to the fire is 2.8°. how far away from the base of the tower is the fire? assume the ground from the tower to the fire is completely flat. round your answer to one decimal place.

Explanation:

Step1: Understand the problem

We have a right triangle where the height of the tower (opposite side to the angle of depression) is 229 feet, and we need to find the distance from the base of the tower to the fire (adjacent side). The angle of depression is equal to the angle of elevation from the fire to the ranger, so we can use the tangent function. The tangent of an angle in a right triangle is defined as $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Let $x$ be the distance from the base of the tower to the fire. So $\tan(2.8^{\circ})=\frac{229}{x}$.

Step2: Solve for x

We can rearrange the formula to solve for $x$. From $\tan(2.8^{\circ})=\frac{229}{x}$, we get $x = \frac{229}{\tan(2.8^{\circ})}$. First, we need to find the value of $\tan(2.8^{\circ})$. Using a calculator, make sure it is in degree mode. $\tan(2.8^{\circ})\approx0.0489$. Then $x=\frac{229}{0.0489}\approx4683.0$.

Answer:

4683.0