going form R to R+2 in a ballon?

Question

A spherical balloon with radius r inches has volume defined by the function below. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 2 inches.

I took V(r)=4/3pi(r+2)^3 and subtracted it from the orignal formula V(r)=4/3pir^3 now do the 4/3pi cnacel each other out? leaving me with 6r^2+12r+8? or do you just subtract the (r+2)^3 from r^3 leaving you with 8pir^2+16pir+(32/3pi)?

Answer 1

The plan is to subtract V(r) from V(r+2)

V(r+2) - V(r)= 4/3pi (r+2)^3 - 4/3pi r^3

factor out 4/3 pi
V(r+2) - V(r)= 4/3pi [(r+2)^3 - r^3]

V(r+2) - V(r)= 4/3pi [r^3 + 6r^2 +12r + 8 - r^3]
V(r+2) - V(r)= 4/3pi [6r^2 +12r + 8]
V(r+2) - V(r)= 8pi [r^2 +2r + 4/3]