Question

the angle of elevation to a nearby tree from a point on the ground is measured to be 42°. how tall is the tree if the point on the ground is 43 feet from the bottom of the tree? round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Use the tangent function

In a right - triangle (formed by the tree, the ground, and the line of sight), the tangent of an angle of elevation $\theta$ is given by $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 42^{\circ}$, the adjacent side is $43$ feet, and the opposite side is the height of the tree $x$. So, $\tan(42^{\circ})=\frac{x}{43}$.

Step2: Solve for $x$

Multiply both sides of the equation $\tan(42^{\circ})=\frac{x}{43}$ by $43$ to isolate $x$. We get $x = 43\times\tan(42^{\circ})$.
We know that $\tan(42^{\circ})\approx0.9004$. Then $x = 43\times0.9004$.
$x=38.7172$.

Answer:

$38.7$ feet