Rates of Change — Question 2

Question 2

Problem:

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

Question 2 - Solution

We are given the volume of a sphere:

$$V = \frac{4}{3}\pi r^3$$

Differentiate both sides with respect to time $t$:

$$\frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt}$$

We are told that:

$$\frac{dV}{dt} = 100, \quad r = 5$$

Substitute into the equation:

$$100 = 4\pi (5)^2 \cdot \frac{dr}{dt} \Rightarrow 100 = 100\pi \cdot \frac{dr}{dt} \Rightarrow \frac{dr}{dt} = \frac{1}{\pi}$$

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