Sphere Related Rates Calculator
Calculate change in Volume, Area, and Radius instantly
Rate of Change of Radius (dr/dt)
Rate of Change of Volume (dV/dt)
Rate of Change of Surface Area (dA/dt)
Calculated Instantaneous Rates
Change in Radius (dr/dt):
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Change in Volume (dV/dt):
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Change in Surface Area (dA/dt):
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Current Static Properties
Current Total Volume:
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Current Total Surface Area:
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Understanding Related Rates for Spheres
In calculus, "Related Rates" problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. Spheres are one of the most common geometric shapes used in these problems, often representing expanding balloons, forming bubbles, or melting snowballs.
The Core Formulas
To solve related rates problems for a sphere, we rely on the formulas for Volume ($V$) and Surface Area ($A$) and differentiate them with respect to time ($t$) using the Chain Rule.
Volume Formula: V = (4/3)πr³
Derivative: dV/dt = 4πr²(dr/dt)
Surface Area Formula: A = 4πr²
Derivative: dA/dt = 8πr(dr/dt)
Derivative: dV/dt = 4πr²(dr/dt)
Surface Area Formula: A = 4πr²
Derivative: dA/dt = 8πr(dr/dt)
Variable Definitions
- r (Radius): The distance from the center of the sphere to its surface.
- dr/dt: How fast the radius is growing or shrinking (e.g., cm/sec).
- dV/dt: How fast the volume is changing (e.g., cm³/sec).
- dA/dt: How fast the surface area is changing (e.g., cm²/sec).
Common Problem Scenarios
1. The Inflating Balloon:
Air is being pumped into a spherical balloon at a rate of 100 cm³/sec (dV/dt). How fast is the radius increasing (dr/dt) when the diameter is 50 cm? This calculator solves this by allowing you to input the Rate of Volume and current Radius to find the Rate of Radius.
2. The Melting Snowball:
A spherical snowball melts such that its radius decreases at a rate of 0.5 cm/min (dr/dt is negative). You can calculate how fast the volume is vanishing at a specific moment.
How to Use This Calculator
- Enter Current Radius: Input the radius of the sphere at the exact moment you are analyzing.
- Select Known Rate: Choose which variable's rate of change you already know (Radius, Volume, or Area).
- Enter Rate Value: Input the numerical value of that rate. Use a positive number for growth (inflation) and a negative number for shrinkage (deflation/melting).
- Calculate: Click the button to see the derivatives for all three variables simultaneously.