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Math Topic: word problems with related rates

Key Terms: representation, sketch, translation, chain rule, implicit differentiation

In the preceding applet you were able to make observations about the rate of change of distance between two ships given their respective rates of change of position or velocities. Making changes in one ship's speed had an effect on the rate of change in distance between them.

Suppose we have an industrial or naval application involving an expanding spherical object. Furthermore, suppose it is known that the volume of the sphere is increasing at the rate of 10 cubic feet per minute. A typical related rate question might ask the following: How fast is the radius changing when the radius measures 5 feet?

Note: A key observation before we begin is the fact that all quantities are changing with respect to time, t. Thus, all derivatives will be with respect to this t.

Step 1: Sketch and label - Identify all known and unknown quantities; make a sketch and label the quantities.

Step 2: Translate - Translate any explicit or implicit relationships within the problem; this step should result in an equation that relates the changing quantities. Highlight the mathematical representation corresponding to the desired rate of change in the question.

Step 3: Differentiate - Using the chain rule, implicitly differentiate both sides of the equation with respect to time t.

Step 4: Substitute and solve - Substitute all known quantities and rates into the resulting equation from step 3; solve for the required rate of change.

Step 5: Check and Conclude- Translate your numerical solution into a verbal statement; check to see that your answer is reasonable and that no errors have been made.

We now return to our original sample related rate problem and apply the step by step guideline.

Suppose we have an industrial or naval application involving an expanding spherical object. Furthermore, suppose it is known that the volume of the sphere is increasing at the rate of 10 cubic feet per minute. A typical related rate question might ask the following: How fast is the radius changing when the radius measures 5 feet?

Step 1: The known quantities are the radius, labeled r = 5 ft; the rate of change of volume represented by dv/dt = 10 cubic ft per minute; the radius is labeled on the corresponding sketch.

Step 2: The relationship between volume and radius is an implicit relationship in the radius (it's not solved for explicitly). We note that v = 4/3 * p * r³. We also emphasize that the solution to the problem will be the final value of dr/dt.

Step 3: We now differentiate our equation to obtain:

dv/dt = 4* p * r² * dr/dt

Step 4: Substituting known quantities we have the following:

10 = 4*p * (5²) * dr/dt

Solving for dr/dt we have: dr/dt = [10/ (4*p *25)] @ .031831 ft/min

Step 5: We conclude with a verbal statement that the radius is increasing at the rate of approximately .031831 feet per minute. (Note that the positive sign of dr/dt indicates that the radius is increasing. This conclusion was reasonable based on the expanding sphere and the known quantities)

For another site (at Hofstra University) related to this topic click here. 

USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil