A balloon which always remains spherical has a variable radius. Find the rate at which its volume is increasing with respect to its radius when the radius is 7 cm.

Answer

To solve the problem of finding the rate at which the volume of a spherical balloon is increasing with respect to its radius when the radius is 7 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Formula**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

2. **Differentiate the Volume with Respect to Radius**:
We need to find the rate of change of volume with respect to the radius, which is denoted as \( \frac{dV}{dr} \). To find this, we differentiate the volume formula:
\[ \frac{dV}{dr} = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) \]
Using the power rule of differentiation, where \( \frac{d}{dr} (r^n) = n r^{n-1} \), we get:
\[ \frac{dV}{dr} = \frac{4}{3} \pi \cdot 3r^2 \]
Simplifying this, we find:
\[ \frac{dV}{dr} = 4 \pi r^2 \]

3. **Substitute the Given Radius**:
Now, we need to evaluate \( \frac{dV}{dr} \) at \( r = 7 \) cm:
\[ \frac{dV}{dr} \bigg|_{r=7} = 4 \pi (7^2) \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Therefore:
\[ \frac{dV}{dr} \bigg|_{r=7} = 4 \pi \cdot 49 = 196 \pi \]

4. **Calculate the Numerical Value**:
Using \( \pi \approx \frac{22}{7} \):
\[ 196 \pi \approx 196 \cdot \frac{22}{7} \]
Simplifying this:
\[ 196 \cdot \frac{22}{7} = \frac{4312}{7} \approx 616 \text{ cm}^2/\text{cm} \]

5. **Determine the Units**:
The unit of \( \frac{dV}{dr} \) is:
\[ \text{Volume unit (cm}^3\text{)} / \text{Radius unit (cm)} = \text{cm}^2 \]

### Final Answer:
The rate at which the volume of the balloon is increasing with respect to its radius when the radius is 7 cm is:
\[ \frac{dV}{dr} = 616 \text{ cm}^2/\text{cm} \]