A spherical balloon with radius r inches has

Question

volume
V (r) =
4
3
� r3 .
Find the function f(r) that represents the
amount of air required to inflate the balloon
from a radius of r inches to a radius of r + 2
inches.

Answer 1

V1=(4/3)*pi*r^3

(r+2)^3=r^3+3r^2*2+3r*2^2+2^3

(r+2)^3=r^3+6r^2+3r*4+8

(r+2)^3=r^3+6r^2+12r+8

V2=(4/3)*pi*(r+2)^3

f(r)=V2-V1=(4/3)*pi*(r+2)^3-(4/3)*pi*r^3=

(4/3)*pi*[(r+2)^3-r^3]=

(4/3)*pi*(r^3+6r^2+12r+8-r^3)=

(4/3)*pi*(6r^2+12r+8)

f(r)=(4/3)*pi*(6r^2+12r+8)