Is \(\alpha (t)\) an increasing or decreasing function with respect to \(t\)?
Increasing Decreasing
elevation and depression
We have other measurements, besides length, that might be changing in a situation.
\[ \text {rate} = \frac {\Delta \text {angle}}{\Delta \text {time}} \]
Note: \(\Delta \text {angle}\) might be measured in degrees or radians. Calculus will use radians.
0.1 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.
Which expression represents \(\sin (\alpha )\)?
\(\frac {F}{H}\) \(\frac {D}{F}\) \(\frac {F}{D}\) \(\frac {H}{D}\)
Which expression represents \(\cos (\beta )\)?
\(\frac {F}{H}\) \(\frac {D}{F}\) \(\frac {F}{D}\) \(\frac {H}{D}\)
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