DATA 605 - Week 14 Exercise

Section 4.2 Exercise 4

A circular balloon is inflated with air flowing at a rate of 10cm\(^3\)/s. How fast is the radius of the balloon increasing when the radius is (a) 1cm? (b) 10cm? (c) 100cm?

Start by using an equation that relates the volume of the balloon to its radius:

\[ V = \frac{4}{3}\pi r^3 \] Differentiate: \[ \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} \] Re-arrange for the rate of interest: \[ \frac{dr}{dt} = \frac{1}{4\pi r^2} \frac{dV}{dt} \] Encapsulate in an R function:

rad_rate <- function(vol_rate, radius) {
  return(1/(4*pi*radius^2)*vol_rate)
}

a) r = 1cm

rad_rate(vol_rate=10, radius=1)

## [1] 0.7958

b) r = 10cm

rad_rate(vol_rate=10, radius=10)

## [1] 0.007958

c) r = 100cm

rad_rate(vol_rate=10, radius=100)

## [1] 7.958e-05