# Instant rate of growth of a spherical balloon

I am having trouble with a problem involving a spherical balloon, and I was hoping someone could help me out. The problem is as follows:

"Air is blown into a spherical balloon so that its volume increases at a rate of $$30 cm^3/s$$. How fast is the radius increasing at the moment when it is 18 cm? Give your answer to an appropriate number of significant figures."

Here is what I have done so far:

I know that the formula for the volume of a sphere is $$V = (4/3)πr^3$$, so I differentiated it with respect to time to get $$dV/dt = 4πr^2(dr/dt)$$, where $$r$$ is the radius and $$dr/dt$$ is the rate at which the radius is changing.

Since the problem gives me the rate of change of volume, $$dV/dt = 30 cm^3/s$$, and I know that the radius at the moment when the volume is increasing at this rate is $$18 cm$$, I can solve for $$dr/dt$$.

However, when I plug in the values and solve for $$dr/dt$$, I am not getting the correct answer.

• Why don't you include your answer? Commented Mar 10, 2023 at 18:22
• Your procedure is correct so you probably just made a mistake. The answer key could also be wrong. Commented Mar 10, 2023 at 18:27

$$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$$
$$\implies 30=4\pi(324)\frac{dr}{dt}$$
$$\implies \frac{dr}{dt}=\frac{5}{216\pi} \text{cm}/\text{s}$$