Question

1. a hiker looks up at a 40 ft tall tree, and he determines that the angle from the ground to the top of the tree is 34°. how far is he from the tree?

Explanation:

Step1: Use the tangent function

In a right - triangle (where the tree height is the opposite side and the distance from the tree is the adjacent side with respect to the given angle), the tangent of an angle $\theta$ is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the distance from the hiker to the tree be $x$. We know that the height of the tree (opposite side) $y = 40$ ft and $\theta=34^{\circ}$. So, $\tan(34^{\circ})=\frac{40}{x}$.

Step2: Solve for $x$

We can rewrite the equation from Step 1 as $x=\frac{40}{\tan(34^{\circ})}$. Since $\tan(34^{\circ})\approx0.6745$, then $x=\frac{40}{0.6745}\approx59.3$ ft.

Answer:

The hiker is approximately \(59.3\) ft from the tree.