What Makes These Related-Rates Problems Different?
Many related-rates problems ask you to track how the volume of a three-dimensional object changes as one or more of its dimensions change. The calculus is usually straightforward. The difficult part is building the correct equation before differentiating.
For cones and spheres, the most important habit is to start with the geometric formula for volume, then differentiate that equation with respect to time. The variables are changing, so quantities such as and are functions of time even when the problem does not explicitly write them that way.
Once the volume equation is written, use implicit differentiation with respect to time. Every changing variable produces its own rate, such as .
The Two Volume Formulas You Need
Volume of a Cone
For a cone with radius and height , the volume is:
Notice that both and may change over time. If the problem gives you a relationship between them, such as a fixed ratio between radius and height, you should use that relationship before or during differentiation.
Volume of a Sphere
For a sphere with radius :
A sphere is often simpler than a cone because there is only one changing geometric variable: the radius.
The Basic Related-Rates Strategy
- Identify the changing quantities. Decide which dimensions and rates are given and which rate the problem asks for.
- Write the geometric equation. For a cone, use . For a sphere, use .
- Differentiate with respect to time. Treat every changing variable as a function of .
- Substitute known rates and dimensions. Use the values that apply at the instant specified in the problem.
- Solve for the requested rate. Keep track of units throughout the calculation.
Do not substitute numerical values for the changing variables before differentiating unless the quantity is genuinely constant. Differentiate the general relationship first, then evaluate it at the requested instant.
Worked Examples
Water is poured into a spherical tank at a rate of . How quickly is the radius increasing when the radius is ?
Water is poured into a conical tank at a rate of . The tank has a height of and a radius of when full. Because the cone maintains its shape as it fills, the radius and height remain proportional. How quickly is the water level rising when the water is high and the radius is ?
A spherical balloon is expanding so that its radius increases at . How quickly is its volume increasing when the radius is ?
The radius of a conical pile of sand increases at , while its height increases at . At a particular instant, the radius is and the height is . How quickly is the volume increasing at that instant?
Why Similar Triangles Matter for Cones
A common AP Calculus cone problem gives you a cone whose shape remains fixed as it fills or empties. In that situation, the radius and height are not independent. Their ratio stays constant.
If a cone has full dimensions and , then similar triangles give:
This lets you replace one changing variable with the other. For example:
After that substitution, the volume can often be written entirely in terms of . The resulting related-rates equation becomes much easier to differentiate.
Growing vs. Shrinking Volumes
The sign of a rate matters. If a sphere is expanding, then is positive. If it is shrinking, then is negative.
For example, if the radius of a spherical balloon is decreasing at , then:
The negative sign tells you that the volume is decreasing. Do not automatically report only the magnitude of the answer; the sign communicates the direction of change.
Common Mistakes
If you plug in the radius or height too early, you can accidentally turn a changing quantity into a constant. Write and differentiate the general volume equation first, then substitute the instantaneous values.
When differentiating with respect to time, remember that . Thus, differentiating gives , not merely .
If a cone fills while maintaining the same shape, its radius and height must remain proportional. Use similar triangles to establish the relationship before solving the rate problem.
Volume rates have cubic units per unit time, such as . Radius and height rates have linear units per unit time, such as . Always check that your final units match the quantity being asked for.
A Fast AP Exam Checklist
- Draw or visualize the solid. Label the changing dimensions.
- Write the volume formula.
- Look for a geometric relationship. For cones, check for similar triangles.
- Differentiate with respect to time.
- Substitute the instantaneous values and rates.
- Solve for the requested derivative.
- Include units and interpret the sign.
Practice Problems
A spherical balloon is inflated so that its radius increases at . How quickly is its volume increasing when the radius is ?
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Water flows into a spherical tank at . How quickly is the radius increasing when the radius is ?
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A conical tank is high and has radius at the top. Water enters at . How quickly is the water level rising when the water is high?
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A conical pile has radius and height at a certain instant. The radius is increasing at and the height is increasing at . How quickly is the volume increasing?
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