11DIFFERENTIATION - Department of Mathematics

766 11 DIFFERENTIATION
Solving Related Rates
Problems
EXAMPLE 8
FIGURE 11.17
The rate at which x is changing with respect
to time is related to the rate of change of
y with respect to time.
1. Assign a variable to each quantity. Draw a diagram if needed.
2. Write the given values of the variables and their rates of change with respect
to t.
3. Find an equation giving the relationship between the variables.
4. Differentiate both sides of this equation implicitly with respect to t.
5. Replace the variables and their derivatives by the numerical data found in
step 2 and solve the equation for the required rate of change.
At a distance of 4000 feet from the launch site, a spectator is observing a
rocket being launched. If the rocket lifts off vertically and is rising at a speed
of 600 feet/second when it is at an altitude of 3000 feet, how fast is the distance
between the rocket and the spectator changing at that instant?
SOLUTION ✔ Step 1 Let
y the altitude of the rocket
x the distance between the rocket and the spectator
at any time t (Figure 11.17).
x
4000 ft
Rocket
Step 2 We are given that at a certain instant of time
y 3000 and
dy
600
dt
Step 3
and are asked to find dx/dt at that instant.
Applying the Pythagorean theorem to the right triangle in Figure
11.17, we find that
x2 y2 40002 Therefore, when y 3000,
y
Launching pad
x 3000 2 4000 2 5000