Helium is pumped into a spherical balloon at a rate of 6 cubic feet per second. How fast is the radius increasing (in ft/s) after 1 minutes?
Note: the volume of a sphere is given by V=4/3pi r^3

To solve the problem, we'll use related rates. We're given that helium is being pumped into a spherical balloon at a rate of 6 cubic feet per second, and we need to find how fast the radius of the balloon is increasing after 1 minute.

### Step 1: Given Information
- The rate at which helium is pumped into the balloon is \( \frac{dV}{dt} = 6 \) cubic feet per second.
- The volume of a sphere is given by the formula: 
  \[
  V = \frac{4}{3} \pi r^3
  \]
- We need to find the rate at which the radius is increasing, \( \frac{dr}{dt} \), after 1 minute.

### Step 2: Differentiate the Volume Formula with Respect to Time
Differentiate both sides of the volume equation with respect to time \( t \):

\[
\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}
\]

### Step 3: Find the Radius After 1 Minute
We need to determine the radius of the balloon after 1 minute. Since the volume of helium pumped is 6 cubic feet per second, after 1 minute (60 seconds), the total volume is:

\[
V = 6 \times 60 = 360 \text{ cubic feet}
\]

Set this equal to the volume formula:

\[
\frac{4}{3} \pi r^3 = 360
\]

Solve for \( r \):

\[
r^3 = \frac{360 \times 3}{4\pi} = \frac{1080}{4\pi} = \frac{270}{\pi}
\]

\[
r = \left(\frac{270}{\pi}\right)^{\frac{1}{3}}
\]

### Step 4: Substitute Values into the Related Rates Equation
Now, substitute the values \( r \) and \( \frac{dV}{dt} = 6 \) into the differentiated equation:

\[
6 = 4 \pi \left(\frac{270}{\pi}\right)^{\frac{2}{3}} \frac{dr}{dt}
\]

Solve for \( \frac{dr}{dt} \):

\[
\frac{dr}{dt} = \frac{6}{4 \pi \left(\frac{270}{\pi}\right)^{\frac{2}{3}}}
\]

Simplify:

\[
\frac{dr}{dt} = \frac{3}{2 \pi \left(\frac{270}{\pi}\right)^{\frac{2}{3}}}
\]

To further simplify, you can compute the value numerically:

\[
r \approx \left(\frac{270}{\pi}\right)^{\frac{1}{3}} \approx 3.65 \text{ ft}
\]

Then,

\[
\

Explore how to solve a related rates problem involving a spherical helium balloon. Understand the relationship between the rate of helium inflow, balloon volume, and the rate of change of its radius after 1 minute. Suitable for advanced high school students studying calculus.

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