How to Approach Related Rates
Related-rates problems describe a situation in which two or more quantities change with time. The key idea is that the quantities are connected by an equation, so their rates of change are connected as well.
For AP Calculus problems, the most reliable strategy is to work symbolically first. Do not substitute the numerical values until after you differentiate the relationship with respect to time.
The Five-Step Method
- Define the variables. Decide what each variable represents and identify the rates you know and the rate you need.
- Write an equation relating the variables. Use the geometry or physical relationship described in the problem.
- Differentiate with respect to time. Every changing variable becomes a time derivative.
- Substitute the known values. Use the information corresponding to the exact instant described in the problem.
- State the answer with units and interpretation. A related-rates answer should describe what is increasing or decreasing and at what rate.
Do not plug numbers into the original equation and then differentiate. The variables represent changing quantities, so you need to differentiate the relationship before evaluating it at the specified instant.
Worked Examples
The radius of a circular oil spill is increasing at a rate of 3 meters per minute. How quickly is the area increasing when the radius is 10 meters?
Water is poured into a cylindrical tank at a rate of 12 cubic feet per minute. The tank has a radius of 4 feet. How quickly is the water level rising?
A 13-foot ladder leans against a vertical wall. The bottom of the ladder is moving away from the wall at 2 feet per second. How quickly is the top of the ladder moving downward when the bottom is 5 feet from the wall?
The length of a rectangle is increasing at 5 centimeters per second while its width is increasing at 2 centimeters per second. How quickly is the area increasing when the length is 12 centimeters and the width is 7 centimeters?
Common Mistakes
The relationship between the variables must first be differentiated with respect to time. Substituting values too early can eliminate the changing variables you need in the derivative equation.
When differentiating a quantity such as , remember that is itself changing with time. The derivative is , not simply .
A quantity moving downward or decreasing can have a negative rate. Keep the sign during the calculation and interpret it at the end.
Rates have units. An answer such as 4.5 is incomplete if the problem asks how quickly an area, volume, distance, or height is changing.
Practice Problems
Try each problem before opening the solution. For every problem, identify the variables, write the relationship, differentiate with respect to time, substitute the values, and include appropriate units.
The radius of a circular pond is increasing at 0.5 meters per minute. How quickly is the area increasing when the radius is 8 meters?
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The radius of a spherical balloon is increasing at 0.2 centimeters per second. How quickly is the volume increasing when the radius is 6 centimeters?
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A 10-foot ladder rests against a wall. The bottom is pulled away from the wall at 1.5 feet per second. How quickly is the top moving downward when the bottom is 6 feet from the wall?
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Water is poured into a conical tank at 5 cubic feet per minute. The tank has height 12 feet and radius 4 feet. How quickly is the water level rising when the water is 3 feet deep?
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The length of a rectangle increases at 3 centimeters per second while the width decreases at 1 centimeter per second. How quickly is the area changing when the length is 10 centimeters and the width is 6 centimeters?
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A person walks directly away from a lamppost at 4 feet per second. The lamppost is 12 feet tall. How quickly is the angle of elevation changing when the person is 5 feet from the base of the lamppost?
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One car travels north at 40 miles per hour while another travels east at 30 miles per hour. Both cars start at the same intersection. How quickly is the distance between them changing 2 hours after they leave?
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A particle moves along a curve satisfying . At the instant when , the particle has . If , determine .
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How to Check Your Work
Before moving on, check four things:
- Equation: Did you choose a relationship that actually connects the quantities?
- Differentiation: Did every changing variable receive a time derivative?
- Substitution: Did you use the values corresponding to the exact instant in the question?
- Units and sign: Does your final rate have the correct units, and does its sign match the situation?
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