Related Rates: Sphere Expansion Calculator
This tool calculates the rate of change of volume (dV/dt) for a sphere based on its current radius and the rate at which the radius is changing (dr/dt).
Calculation Results:
Understanding Related Rates in Calculus
Related rates problems involve finding the rate at which one quantity changes by relating that quantity to other quantities whose rates of change are already known. These problems are a fundamental application of implicit differentiation with respect to time (t).
The Physics and Math Behind Spherical Expansion
When you blow air into a spherical balloon, both the radius (r) and the volume (V) change over time. Because the volume of a sphere is strictly dependent on its radius, their rates of change are mathematically linked.
Formula: V = (4/3)πr³
Differentiating with respect to time (t):
dV/dt = 4πr² (dr/dt)
Step-by-Step Calculation Example
Imagine a snowball melting (contracting) such that its radius decreases at a constant rate of 0.5 cm per minute. If the radius is currently 10 cm, how fast is the volume decreasing?
- Step 1: Identify given values. r = 10, dr/dt = -0.5 (negative because it's decreasing).
- Step 2: Use the derivative formula. dV/dt = 4π(10)²(-0.5).
- Step 3: Solve. dV/dt = 4π(100)(-0.5) = -200π.
- Step 4: Final Result. The volume is decreasing at approximately 628.32 cubic centimeters per minute.
Key Variables Defined
| Variable | Description |
|---|---|
| r | The instantaneous radius of the object at a specific moment. |
| dr/dt | How fast the radius is growing or shrinking (units per time). |
| dV/dt | The resulting rate of change for the total volume. |