Question

A spherical balloon is inflated with gas at a rate of 500 cubic centimeters per minute. (a) How fast is the radius of the balloon changing at the instant the radius is 40 centimeters

Answer 1

`\mathbf{ (dr)/(dt) = 0.03730 \ cm/min}`

Explanation:

The rate of the inflation of the balloon with time can be denoted as:

`(dv)/(dt) = 500 \ cm^3/m`

To determine; how fast does the radius change with time.

i.e.

`(dr)/(dt)=???`

where r = 40 cm and the volume of sphere =
`(4)/(3) \pi r^2`

∴

`(dv)/(dt)= (4)/(3) \pi 2(r^3) (dr)/(dt)`

`500= (4)/(3) \pi * 2(40^2) (dr)/(dt)`

`500= 13404.13 (dr)/(dt)`

`(500)/(13404.13) = (dr)/(dt)`

`\mathbf{ (dr)/(dt) = 0.03730 \ cm/min}`