A spherical balloon with radius r inches has

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.

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)

To 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, we need to find the difference in volume between the two radii.

Let's break down the problem step by step:

1. Find the volume of the balloon with radius r inches: V(r) = (4/3)πr^3

2. Find the volume of the balloon with radius (r + 2) inches: V(r + 2) = (4/3)π(r + 2)^3

3. Calculate the difference in volume between the two radii:
ΔV(r) = V(r + 2) - V(r) = (4/3)π(r + 2)^3 - (4/3)πr^3

4. Simplify the expression:
ΔV(r) = (4/3)π(r^3 + 6r^2 + 12r + 8) - (4/3)πr^3
= (4/3)π(6r^2 + 12r + 8)

Therefore, 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 is:
f(r) = (4/3)π(6r^2 + 12r + 8)