what is the answer

Question

A spherical balloon is inflated with gas at a rate of 900 cubic centimeters per minute.

(a) How fast is the radius of the balloon changing at the instant the radius is 50 centimeters?

 

  cm/min

(b) How fast is the radius of the balloon changing at the instant the radius is 90 centimeters?

 

 cm/min

Answer 1

Here we need to apply related rates.

the volume of a sphere is:

V=(4/3)pi*r^3

the hint here is that the balloon is changing volume with respect to time, so that tells us we need to take the time derivative of the volume equation with respect to time.

Once doing that, we get:

dV/dt = (4*pi*r^2) dr/dt

I don't want to fully solve this question for you so I'll leave it here.

Another hint: they give you some of the rates you need to plug in in order to find items "a" and "b".

Hope this helps!

Answer 2

Volume of a sphere = 4/3 pi(r^3)

V'(r) = 900 = (4/3)(3)pir^2(r') = 4(pi)(50)^2(r')

r' = 900/4pi(2500)

r' = .02864 centimeters per second'

r' = about 0.029 cm/s when radius = 50 cm and V'=900 cm^3/sec

when r =90 cm and V'=900 cm^3/sec

r' = 900/4pi(90^2) = 900/4pi(8100)

r' =about 0.008842 cm/sec