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