Question

the angle of elevation to a nearby tree from a point on the ground is measured to be 63°. how tall is the tree if the point on the ground is 71 feet from the bottom of the tree? round your answer to the nearest hundredth 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, $\theta = 63^{\circ}$, the adjacent side $a = 71$ feet, and the opposite side (height of the tree) is $x$. So, $\tan63^{\circ}=\frac{x}{71}$.

Step2: Solve for $x$

We know that $\tan63^{\circ}\approx1.9626$. Then, from the equation $x = 71\times\tan63^{\circ}$, substitute the value of $\tan63^{\circ}$. So, $x=71\times1.9626$.

Step3: Calculate the value of $x$

$x = 71\times1.9626=139.3446$.

Answer:

$139.34$ feet