Question

1. a ship in distress fires a signal flare. the flare explodes at a height of 800 feet above the ship. a rescue ship sees the flare and measures the angle of elevation from its location up to the flare to be about 16.0°. about how far is the ship that is in distress from the rescue ship?

Explanation:

Step1: Set up the right - triangle model

We have a right - triangle where the height of the flare above the ship in distress (opposite side) is $h = 800$ feet and the angle of elevation $\theta=16.0^{\circ}$. We want to find the distance $d$ (adjacent side) between the rescue ship and the ship in distress. We use the tangent function in a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.
$\tan\theta=\frac{h}{d}$

Step2: Solve for $d$

We can re - arrange the formula $\tan\theta=\frac{h}{d}$ to solve for $d$. So, $d = \frac{h}{\tan\theta}$.
Substitute $h = 800$ feet and $\theta = 16.0^{\circ}$ into the formula. Since $\tan(16.0^{\circ})\approx0.2867$, then $d=\frac{800}{\tan(16.0^{\circ})}=\frac{800}{0.2867}\approx2790$ feet.

Answer:

About 2790 feet