Solve common related rates problems involving circles, spheres, and Pythagorean geometry.
Expanding Circle (Area)
Expanding Sphere (Volume)
Leaning Ladder (Pythagorean Theorem)
Calculated Rate:
Enter values and click calculate
Understanding Related Rates in Calculus
Related rates are problems in calculus where we find the rate at which one quantity changes by relating it to other quantities whose rates of change are already known. This is a direct application of the Chain Rule. Usually, we differentiate both sides of an equation with respect to time (t).
The Core Formulas
Circles: If the area is A = πr², then the rate of change of area is dA/dt = 2πr(dr/dt).
Spheres: If the volume is V = (4/3)πr³, then the rate of change of volume is dV/dt = 4πr²(dr/dt).
Pythagorean (Ladder): For x² + y² = L², where the ladder length L is constant, the derivative is 2x(dx/dt) + 2y(dy/dt) = 0, which simplifies to dy/dt = -(x * dx/dt) / y.
Step-by-Step Solving Guide
To solve any related rates problem manually, follow these steps:
Identify the variables: Determine what is changing and what is constant.
Find the equation: Write a formula that relates the variables (e.g., Volume, Area, or Pythagorean theorem).
Differentiate: Use implicit differentiation with respect to time (t).
Substitute: Plug in the known values for the specific moment in time requested.
Solve: Isolate the target rate (e.g., dA/dt or dy/dt).
Example Calculation
Suppose a ladder 10 feet long is leaning against a wall. The bottom of the ladder is 6 feet from the wall (x = 6) and is being pushed toward the wall at 2 ft/s (dx/dt = -2). How fast is the top of the ladder moving up the wall?