Calculus

Question 3

Water flows onto a flat surface at a rate of 5 cm^3/s forming a circular puddle 10mm deep. How fast is the radius growing when the radius is:

volume of puddle = area of circle x depth depth = 10mm = 1cm

\[\frac{dV}{dt} = \pi 2r \frac{dr}{dt} h\] \[\frac{dr}{dt} = \frac{\frac{dV}{dt}}{2 \pi rh}\]

10 cm

## [1] "The radius change over time (cm/s) when the radius is 10cm is:"
## [1] 0.07957747

100 cm

## [1] "The radius change over time (cm/s) when the radius is 100cm is:"
## [1] 0.007957747

Question 4

A circular balloon is inflated with air flowing at a rate of 10 cm^3/s. How fast is the radius of the balloon increasing when the radius is:

\[V = \frac{4}{3} \pi r^3\] \[\frac{dV}{dt} = 4 \pi r ^2 \frac{dr}{dt}\] \[\frac{dr}{dt} = \frac{\frac{dV}{dt}}{4 \pi r^2}\]

10 cm

## [1] "The radius change over time (cm/s) when the radius is 10cm is:"
## [1] 0.007957747

100 cm

## [1] "The radius change over time (cm/s) when the radius is 100cm is:"
## [1] 7.957747e-05