Find fcn. representing volume of air needed to inflate....

[jenn9580]

I'm stuck! I need to find a function that represents the amount of air required to inflate a balloon from a radius of r inches to a radius of r+3 inches.

I know that a balloon is a sphere, therefore the volume is V(r)=4/3pir^3 & if I take the derivative of that function, I get 4pir^2. I'm just not sure how to find the function I need.

Thanks so much!

 

[soroban]

Re: Calculus Function Question

Hello, Jenn!

I need to find a function that represents the amount of air required
to inflate a balloon from a radius of \(\displaystyle r\) inches to a radius of \(\displaystyle r+3\) inches.
I know that a balloon is a sphere, therefore the volume is: \(\displaystyle V(r)\:=\:\frac{4}{3}\pi r^3\;\) . . . Good!

They want the actual change in volume . . . not the rate of change of volume.
. . So we don't need the derivative.

This is just an "arithemtic" problem.

The original volume is: \(\displaystyle \:V_1\:=\:\frac{4}{3}\pi r^3\) in³.
The larger volume is: \(\displaystyle \:V_2\:=\:\frac{4}{3}\pi(r\,+\,3)^3\) in³

The difference is: \(\displaystyle \:\Delta V \;\;=\;\;\frac{4}{3}\pi(r\,+\,3)^3\,-\,\frac{4}{3}\pi r^3 \;\;=\;\;\frac{4}{3}\pi\left[(r\,+\,3)^3\,-\,r^3\right]\)

. . \(\displaystyle = \;\;\frac{4}{3}\pi\left[r^3\,+\,9r^2\,+\,27r\,+\,27\,-\,r^3\right] \;\;=\;\;\frac{4}{3}\pi\left[9r^2\,+\,27r\,+\,27\right]\)


Therefore: \(\displaystyle \:\Delta V \;=\;12\pi\left(r^2\,+\,3r\,+\,3\right)\,\) in³.

 

[jenn9580]

thank you! i was totally going in the wrong direction! it makes perfect sense now