Air is escaping from a spherical balloon at the rate of 2𝑐𝑚3per minute. How fast is the radius shrinking when the volume is 36𝜋𝑐𝑚3?

Answer 1

This is a classic related-rates problem.

First, we see the information states the volume (due to the dimensions of the information) is decreasing (means negative) at 2 cm³. Therefore we can write this using calculus notation as ...

dV/dt = -2 cm³

We are asked to find how fast the radius is shrinking. Using the volume of a sphere...

V = 4/3 * π*r³

we can compute the derivative of this formula with respect to time (t).

d/dt r³ = 3 * r² * dr/dt

so...

dV/dt = 4/3 * π * 3 * r² * dr/dt

-2 = 4*π * r² * dr/dt

-1/(2π) = r² * dr/dt

-1/(2πr²) = dr/dt

Now we need to find the radius to determine dr/dt. We know the volume at the particular point is 36*π cm³.

Therefore --> 36π = 4/3 * π *r³

Solving for r yields --> (3*36/4)^1/3 = r --> 3

Now we compute dr/dt = -1/(2π*3²) = -1/(18π)