A spherical balloon with radius r inches has volume V(r) = 4 3 πr3. Find an expression that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 4 inches. (Express your answer in terms of π and r.)

Question

A spherical balloon with radius r inches has volume  V(r) =  4 3 πr3.  Find an expression that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 4 inches. (Express your answer in terms of π and r.)

campbell_st · Answer

for the larger radius $$V(r + 4) = 4/3 \pi (r + 4)^3 $$

the smaller radius $$V(r) = 4/3 \pi r^3$$

the increase in volume when the radius increases from r to r + 4
is 

larger radius volume - smaller radius volume

$$V = 4/3 \pi[(r + 4)^3 - r^3]$$

just expand and simplify

anonymous · Answer

so the expanded version would be V=4/3pi(r^3+64)-r^3 ?

anonymous · Answer

Your expansion of (r+4)^3 isn't correct.  Multiply it out step by step.  Start with
(r+4)(r+4)(r+4) and multiply step by step.

anonymous · Answer

so i got r^3+12r^2+48r+64. is that right? what do i do now?

anonymous · Answer

Yup.  Put that in, gather the like terms, and you will have your answer.

anonymous · Answer

I think you might get something similar to$$V=\frac{4 \pi}{3}(12r^2+48r+64)=\frac{16 \pi}{3}(3r^2+16r+16)$$

anonymous · Answer

OOOPS!!  Bad factoring on my previous answer!  Correction:

$$V=\frac{4 \pi}{3}(12r^2+48r+64)=\frac{16 \pi}{3}(3r^2+12r+16)$$Sorry for the careless goof.

Do math every day.

anonymous · Answer

that's the answer i got! thank you so much for your help! (:

anonymous · Answer

No sweat.  Do math every day.