Question

Helium is pumped into a spherical balloon at a constant rate of 3 cubic feet per second. How fast is the radius increasing after 4 minutes? At what time (if any) is the radius increasing at a rate of 140 feet per second?

Answer 1

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Volume equation is V = 4pi r^3 / 3

we need to implicitly differentiate this equation

if we did we get: dV/dt = 4pi r^2 dr/dt

We need dr/dt in this problem, but we don;t know r either

So we use the fact that volume is changing at constant rate

that is 3 / 1 = V/ (4x60) see that I used 4 min that give and convert it to seconds 4x60

that is V= 720 cubic ft

now we use this in the first equation 120 = 4 pi r^3/3 we get r = 3th root (90/pi)= 3.0598

going back to formula we obtain from differentiation

we plug in all the pieces.

3 = 4pi (3.0598)^2 dr/dt solve for dr/dt

dr/dt = 0.0254 ft/sec