Rates of Change — Question 4

Question 4

Problem:

A spherical balloon is being inflated so that its volume increases at a rate of $100 \, \text{cm}^3/\text{s}$. How fast is the radius increasing when the radius is $5 \, \text{cm}$?

Question 4 - Solution

We use the formula for the volume of a sphere: $$V = \frac{4}{3} \pi r^3$$

Differentiate both sides with respect to $t$: $$\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$$

Substitute: $$\frac{dV}{dt} = 100, \quad r = 5$$ $$100 = 4\pi (5)^2 \frac{dr}{dt} \Rightarrow 100 = 100\pi \frac{dr}{dt} \Rightarrow \frac{dr}{dt} = \frac{100}{100\pi} = \frac{1}{\pi}$$

Answer: $$\boxed{\frac{dr}{dt} = \frac{1}{\pi} \text{ cm/sec}}$$