How Fast Does a Balloon's Radius Expand When Inflated?

[Monoxdifly]

Icha is going to blow a sphere-shaped rubber balloon. She uses a pump to infuse the air with the volume addition rate \(\displaystyle 40cm^3/s\). If the radius addition rate is 20 cm/s, the radius of the sphere after being blown is ...
A. \(\displaystyle \frac{1}{\sqrt\pi}\) cm
B. \(\displaystyle \frac{1}{\sqrt{2\pi}}\) cm
C. \(\displaystyle \frac{1}{2\sqrt\pi}\) cm
D. \(\displaystyle \frac{1}{3\sqrt\pi}\) cm
E. \(\displaystyle \pi\) cm

Please give me some hints. Thanks.

 

[skeeter]

Icha is going to blow a sphere-shaped rubber balloon. She uses a pump to infuse the air with the volume addition rate \(\displaystyle 40cm^3/s\). If the radius addition rate is 20 cm/s, the radius of the sphere after being blown is ...
A. \(\displaystyle \frac{1}{\sqrt\pi}\) cm
B. \(\displaystyle \frac{1}{\sqrt{2\pi}}\) cm
C. \(\displaystyle \frac{1}{2\sqrt\pi}\) cm
D. \(\displaystyle \frac{1}{3\sqrt\pi}\) cm
E. \(\displaystyle \pi\) cm

Please give me some hints. Thanks.

take the derivative of the sphere volume formula w/respect to time ...

 

[Monoxdifly]

\(\displaystyle 40cm^3/s=4\pi r^2(20cm/s)\)
\(\displaystyle 2cm^2=4\pi r^2\)
\(\displaystyle r^2=\frac{1cm^2}{2\pi}\)
\(\displaystyle r=\frac{1}{\sqrt{2\pi}}cm\)?

 

[MarkFL]

For a sphere of radius $r$, the volume is:

\(\displaystyle V=\frac{4}{3}\pi r^3\)

Differentiating w.r.t time $t$, we have:

\(\displaystyle \d{V}{t}=4\pi r^2\d{r}{t}\implies r=\frac{1}{2}\sqrt{\frac{\d{V}{t}}{\pi\d{r}{t}}}\)

Plugging in the given numbers, we have:

\(\displaystyle r=\frac{1}{2}\sqrt{\frac{40\frac{\text{cm}^3}{\text{s}}}{20\pi\frac{\text{cm}}{\text{s}}}}=\frac{1}{\sqrt{2\pi}}\text{ cm}\quad\checkmark\)

I will move this thread to our Calculus forum. :)

 

[Monoxdifly]

I will move this thread to our Calculus forum. :)

I thought this was pre-calculus because it was a question in high school graduation exam.

 

[MarkFL]

I thought this was pre-calculus because it was a question in high school graduation exam.

It is a related rate problem involving differential calculus (so that's why I moved it). Many places teach elementary calculus in high school. :)

 

[Monoxdifly]

What is the actual limit between pre-calculus and calculus, anyway?