$$\text {rate} = \frac {\Delta \text {angle}}{\Delta \text {time}}$$

Note: $\Delta \text {angle}$ might be measured in degrees or radians. Calculus will use radians.

Airplane

An airplane is flying overhead on a level flight path, 5 miles above the ground. The plane is travelling at a constant speed and will travel directly over a tracking station. The tracking station’s radar antennea measures the distance from the station to the plane. If the distance between the station and the plane is decreasing at a rate of $350$ miles per hour when that distance is $10$ miles, then what is the speed of the plane?

Step 1: A Picture

Step 2: Identify Measurements

$\blacktriangleright$ We have two pertinent angles:

Angles $\alpha$ and $\beta$ both are functions of $t$: $\alpha (t)$ and $\beta (t)$.

• $\alpha$ is the angle of elevation from the station to the plane.
• $\beta$ is the angle of depression from the plane to the station.

These angles are related to the sides of the triangle through sine and cosine.