Question

the angle of elevation to a nearby tree from a point on the ground is measured to be 31°. how tall is the tree if the point on the ground is 88 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, $\theta = 31^{\circ}$, the adjacent side (distance from the point on the ground to the bottom of the tree) is $88$ feet, and the opposite side (height of the tree) is $x$. So, $\tan(31^{\circ})=\frac{x}{88}$.

Step2: Solve for $x$

We know that $\tan(31^{\circ})\approx0.60086$. Then, from $x = 88\times\tan(31^{\circ})$, substituting the value of $\tan(31^{\circ})$, we get $x=88\times0.60086$.
$x = 52.87568$.

Answer:

$52.9$ feet