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Related Rates: Inflating Sphere

Also known as balloon related rates · dV/dt from dr/dt · how fast is the radius growing · related rates sphere · chain rule volume rate

dVdt=4πr2 drdt\frac{dV}{dt} = 4\pi r^{2}\,\frac{dr}{dt}

Worked example: balloon at 100 cm³/s, r = 25 cm → dr/dt = 1.273e-4 m/s — press Try an example to run it live, then adjust anything.

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See also
Sphere Volume — The V = 4/3 πr³ this page differentiates, when you want the volume itself rather than how fast it is changing.Sphere Surface Area — The 4πr² in front is not a coincidence: it is the sphere's own surface area, which is what a shell of new volume spreads across.Power Rule Derivative — Where the 4πr² came from: the power rule applied to r³ inside the volume formula.

Learning zone

A related-rates problem has two quantities tied by a geometric relation and asks how fast one changes given the other. Differentiate the relation with respect to time and the chain rule supplies the link: from V = 4/3 πr³ comes dV/dt = 4πr² · dr/dt, where 4πr² is nothing other than the sphere's own surface area. That is the physical reading — new volume arrives as a thin shell spread over the whole skin, so the faster the skin grows the more volume it takes, and a big sphere swallows far more air per millimetre of radius than a small one. Worked example: pump air into a spherical balloon at 100 cm³/s and ask how fast the radius is moving when r = 25 cm. The surface is 4π(25)² = 7853.98 cm², so dr/dt = 100 ÷ 7853.98 = 0.01273 cm/s. At r = 5 cm the same pump would be driving the radius out twenty-five times faster.

The order of operations is where these problems are won and lost: differentiate first, substitute the instant's values second. Putting r = 25 into the volume formula before differentiating turns a variable into a constant and gives dV/dt = 0, the classic zero that tells you nothing. It is worth noticing that the same relation is why an oil drop spreads slower and slower, and why a raindrop evaporating at a rate proportional to its surface area loses radius at a constant speed — the 4πr² cancels. The pattern generalises to any shape: differentiate the volume formula, keep the chain-rule factor, substitute at the end.

Related Rates: Inflating Sphere
dVdt=4πr2 drdt\frac{dV}{dt} = 4\pi r^{2}\,\frac{dr}{dt}
Where
  • dVdt\frac{dV}{dt}= Rate of volume change (m³/s)
  • rr= Radius at that instant (m)
  • drdt\frac{dr}{dt}= Rate of radius change (m/s)