Download Section 3.7

AP CALCULUS NOTES
SECTION 3.7 RELATED RATES
In this section we will try to find the rate at which some quantity is changing by relating the
quantity to other quantities whose rates of change are known.
Ex.1.) Suppose we are pumping air into a spherical balloon. In this case, both the volume and
radius of the balloon are increasing over time. If V  volume and r  radius of the balloon at
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an instant of time, then V   r 3 .
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To find the related rates equation, we differentiate.
a.) If the volume is increasing at a rate of 10 ft3/min and the radius is 3 ft, how fast is the
radius increasing at this instant?
STRATEGIES FOR SOLVING RELATED RATES:
1. Define all variables. Use t for time. Assume that all variables are differentiable with
respect to time. Identify the rates of change that are known and the rate of change that
is to be found.
2. Write an equation that relates the variables. Often times you will be able to express
this relationship using an area formula, a volume formula, or the pythagorean theorem.
In some cases, you will need to use a trigonometric function. (Note: In a geometric
problem, you may find it helpful to draw an appropriately labeled figure that illustrates
the relationship involving these quantities.)
3. Differentiate (using implicit differentiation) both sides of the equation with respect to
time.
4. Evaluate the equation by substituting the known values for the quantities and their
rates of change at the moment given.
5. Solve for the value of the remaining rate of change at this moment.
Note: Always perform the differentiation in step 3 before performing the substitution in
step 4.
Ex.2.) How rapidly will the fluid level inside a vertical cylindrical tank drop if we pump the
fluid out at a rate of 3000 L/min for a tank whose radius is 10m ?
Ex.3.) A hot air balloon rising straight up from a level field is tracked by a range finder 500 ft
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from the liftoff point. At the moment the range finder’s elevation angle is , the angle is
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increasing at the rate of .14 rad/min. How fast is the balloon rising at that moment?
Ex.4.) A police cruiser, approaching a right-angled intersection from the north, is chasing a
speeding car that has turned the corner and is now moving straight east as shown below.
When the cruiser is .6 mi north of the intersection and the car is .8 mi to the east, the police
determine with radar that the distance between them and the car is increasing at 20 mph. If
the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car?
Ex.5) Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and
has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the
water is 6 ft deep?