Question

an observer (o) spots a plane (p) taking off from a local airport and flying at a 29° angle horizontal to her line of sight and located directly above her. the observer also notices a bird (b) circling a tower (t). if the distance from the plane (p) to the tower (t) is 6,000 feet, how far is the bird (b) from the plane (p)? round to the nearest whole number.
6,857 feet
10,824 feet
12,371 feet
22,063 feet

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle with an angle of $29^{\circ}$ and the side adjacent to the angle (distance from plane to tower) $PT = 6000$ feet, and we want to find the hypotenuse $BP$. We use the cosine function. $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 29^{\circ}$ and the adjacent side to $\theta$ is the distance from the plane to the tower, and the hypotenuse is the distance from the bird to the plane. So, $\cos(29^{\circ})=\frac{6000}{x}$, where $x$ is the distance from the bird to the plane.

Step2: Solve for $x$

We can re - arrange the formula $\cos(29^{\circ})=\frac{6000}{x}$ to $x=\frac{6000}{\cos(29^{\circ})}$. We know that $\cos(29^{\circ})\approx0.8746$. Then $x=\frac{6000}{0.8746}\approx6857$ feet.

Answer:

6,857 feet