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RELATED RATES APPLICATIONS
Real - World Applications:
Modeling the real world: One of the most fundamental concepts of calculus is the fact that the derivative is used to model change. In related rate applications one tries to find the rate at which one quantity is changing by relating it to other quantities with known rates of change. In the preceding applet we applied related rates to the determination of the rate at which the distance between two ships changes. However, this example is just one of a variety of applications that pertain to related rates. In the following example related rates are applied to jet aircraft and radar tracking. (From Calculus by Freilich and Greenleaf)
- Radar Tracking and Jet Aircraft:
A low-flying jet aircraft covering a straight course is tracked by a radar station set 6 miles to one side of the flight path (see figure below). Here, x = (distance from a reference marker on the course), and s = (direct distance from aircraft to radar unit). A radar unit can measure only the "range" s and the rate of change ds/dt.
Suppose the observed values are s = 10 miles and ds/dt = 800 mph. Calculate the actual speed dx/dt of the aircraft.
Solution: By the Pythagorean Theorem we have : s2 = x2 + 36
Differentiating each side with respect to t using the chain rule, we have
2s ds/dt = 2x dx/dt
and solving this equation for the desired rate , we have
dx/dt = s/x ds/dt.
When s = 10 , the Pythagorean Theorem gives x = 8, and thus, substituting these values and ds/dt = 800,
We obtain dx/dt = 10/8 * 800 = 1000
Our conclusion: The actual speed of the aircraft is 1000 mph.
A short list of additional real world applications include the following:
- Construction for Bancroft Hall: Suppose a metal pipe is leaning against a wall of Bancroft Hall. If the top of the pipe slides down the wall at a given rate, we can determine how fast the bottom of the pipe is moving, using related rates.
- Baseball: Determining a player's rate of change of distance from home plate when he is running from second base to third and his speed and distance from third base is known.
- Rockets: A camera is mounted at a point so many feet from a rocket launching pad. The rocket rises vertically and the elevation of the camera needs to change at just the right rate to keep it in sight. In addition, the camera-to-rocket distance is changing constantly, which means the focusing mechanism will also have to change at just the right rate to keep the picture sharp. Related rates applications can be used to answer the focusing problem as well as the elevation problem.
- Economics: If C= f (x) is a function that relates cost, C, to x, the number of items sold, then both C and x can be considered to be functions of time. Thus, we can answer questions about the rate of change of cost with respect to time or about the rate at which the sales per unit time are changing depending on which items are known. The same model can be applied to profit, demand, supply, and revenue functions.
- More Aircraft Applications:
Suppose one airplane flies over BWI Airport at a given rate, flying east. Twelve minutes later a second plane flies over BWI at its rate, flying north. A typical related rate application would calculate the rate at which they were separating at a later point in time. This is essentially the problem illustrated with boats by the applet in this section.
A second application could calculate the rate at which an airplane is approaching an observer. In this case, the airplane's altitude and speed as it passes directly overhead are known.
USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil