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)