Question

air is being pumped into a spherical balloon at a rate of 100 cm^3/sec. How fast is the diameter increasing when the radius is 5 cm?

Answer 1

Start with volume equation for sphere

`V = (4)/(3) \pi r^3`

Take derivative with respect to time (Implicit differentiation)

`(dV)/(dt) = 4 \pi r^2 (dr)/(dt)`

Sub in rate for Volume of 100 and given radius of 5, solve for dr/dt

`(dr)/(dt) = (100)/(4 \pi 5^2) = (1)/(\pi)`
Finally we have to relate this to rate diameter is changing.
D = 2r

`(dD)/(dt) = 2 (dr)/(dt) = (2)/(\pi)`
Final Answer:
Diameter is increasing at a rate of 2/pi cm/sec