Question

a plane rises from take - off and flies at an angle of 9° with the horizontal runway. when it has gained 500 feet, find the distance, to the nearest foot, the plane has flown. the plane has flown approximately feet. (do not round until the final answer. then round to the nearest foot as needed.)

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the vertical side (opposite side to the given angle) is 500 feet and the angle with the horizontal is 9°. We want to find the hypotenuse (distance the plane has flown). We use the sine function since $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.
Let the hypotenuse be $c$. So, $\sin(9^{\circ})=\frac{500}{c}$.

Step2: Solve for $c$

We can re - arrange the formula $\sin(9^{\circ})=\frac{500}{c}$ to get $c = \frac{500}{\sin(9^{\circ})}$.
We know that $\sin(9^{\circ})\approx0.156434$. Then $c=\frac{500}{0.156434}\approx3196$.

Answer:

3196