Question

emir is standing in a treehouse and looking down at a swingset in the yard next - door. the angle of depression from emirs eyeline to the swingset is 33°69, and emir is 10 feet from the ground. how many feet is the base of the tree from the swingset? round your answer to the nearest foot. 15 feet 18 feet 20 feet 24 feet

Explanation:

Step1: Recall tangent - angle relationship

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. The angle of depression $\theta = 33^{\circ}69'$. First, convert $33^{\circ}69'$ to decimal degrees. Since $69'=\frac{69}{60}=1.15^{\circ}$, then $\theta=33 + 1.15=34.15^{\circ}$. The height of Emir from the ground (opposite side) is $h = 10$ feet, and let the distance from the base of the tree to the swingset be $d$ (adjacent side).

Step2: Set up the tangent equation

$\tan(34.15^{\circ})=\frac{10}{d}$.

Step3: Solve for $d$

$d=\frac{10}{\tan(34.15^{\circ})}$. Using a calculator, $\tan(34.15^{\circ})\approx0.676$. Then $d=\frac{10}{0.676}\approx14.8$. Rounding to the nearest foot, $d = 15$ feet.

Answer:

15 feet