Related Rates Calculator
Solve related rates problems with comprehensive step-by-step calculus solutions and detailed analysis. Our calculus calculator supports balloon, ladder, cone, sphere, and tank problems with implicit differentiation and chain rule applications.
Last updated: December 15, 2024
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Equation:
V = (4/3)πr³
Units: cm
Units: cm³
Enter the rate of change you know (with appropriate sign)
Solution Analysis
Solution:
dV/dt = 628.32 cm³/s
Analysis Type:
Step-by-step solution
Primary Equation:
V = (4/3)πr³
Related Rates Equation:
dV/dt = 4πr² × dr/dt
Step-by-Step Solution:
- 1. Start with volume equation: V = (4/3)πr³
- 2. Differentiate both sides: dV/dt = 4πr² × dr/dt
- 3. Substitute r = 5 cm and dr/dt = 2 cm/s
- 4. Calculate: dV/dt = 4π(5)² × 2 = 628.32 cm³/s
Analysis:
For a spherical balloon with radius 5 cm, when the radius increases at 2 cm/s, the volume increases at 628.32 cm³/s. This demonstrates how small changes in radius lead to much larger changes in volume due to the cubic relationship.
Related Rates Tips:
- • Setup: Identify the geometric relationship between variables
- • Differentiate: Take derivative with respect to time using chain rule
- • Substitute: Plug in known values at the specific moment
- • Check: Verify units and sign make physical sense
Quick Example Result
For a balloon with radius 5 cm expanding at 2 cm/s:
dV/dt = 628.32 cm³/s
Using dV/dt = 4πr² × dr/dt
How This Calculator Works
Our related rates calculator applies the principles of implicit differentiation and the chain rule to solve problems where multiple quantities change with respect to time. The calculator identifies geometric relationships, applies calculus techniques, and provides step-by-step solutions for various real-world scenarios involving changing rates.
Related Rates Solution Process
Establish equation relating variables (geometry/physics)Differentiate both sides with respect to timeUse d/dt[f(x)] = f'(x) × dx/dt for time derivativesPlug in known values and solve for unknown rateThe related rates algorithm systematically applies calculus principles to real-world problems. For example, in balloon problems, the volume V = (4/3)πr³ becomes dV/dt = 4πr² × dr/dt after differentiation. The chain rule ensures that every variable changing with time contributes its rate of change to the final equation.
Mathematical relationships for common related rates problems
Mathematical Foundation
Related rates problems are based on the fundamental calculus principle that if variables are related by an equation, then their rates of change are also related. This connection is established through implicit differentiation, where we differentiate both sides of an equation with respect to time. The chain rule is essential because most variables in these problems are functions of time, requiring us to multiply by the appropriate time derivative.
- Implicit differentiation connects related quantities through their rates of change
- The chain rule ensures proper handling of composite functions involving time
- Geometric relationships provide the foundation equations for most problems
- Sign analysis helps determine whether quantities are increasing or decreasing
Sources & References
- Calculus: Early Transcendentals - James Stewart, Daniel K. Clegg, Saleem WatsonComprehensive coverage of related rates problems and solution techniques
- Paul's Online Math Notes - Related Rates TutorialDetailed tutorial with examples and step-by-step solutions
- Calculus with Analytic Geometry - Howard Anton, Irl C. Bivens, Stephen DavisAdvanced treatment of implicit differentiation and applications
Need help with other calculus concepts? Check out our implicit derivative calculator and derivative calculator.
Get Custom Calculator for Your PlatformExample Analysis
Problem Setup:
- Ladder length: L = 13 feet (constant)
- Bottom distance: x = 5 feet from wall
- Top height: y = 12 feet up the wall
- Bottom rate: dx/dt = 2 ft/s (moving away)
- Find: dy/dt (rate top is sliding down)
Solution Process:
Result: The top of the ladder slides down at 5/6 ≈ 0.83 ft/s
This classic related rates problem demonstrates the Pythagorean relationship between the ladder's position and the wall. When the bottom moves away from the wall at 2 ft/s, the top slides down at approximately 0.83 ft/s. The negative sign indicates downward motion, which makes physical sense. Notice that the top moves slower than the bottom because it's farther from the pivot point (the corner where wall meets ground). The rates are inversely related to the distances: since x = 5 and y = 12, and dx/dt = 2, we get dy/dt = -(5/12) × 2 = -5/6 ft/s.
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