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Let's solve the problem step by step.

### Given Information:
- The balloon is being deflated at a rate of \( \frac{dV}{dt} = -5 \, \text{in}^3/\text{sec} \).
- The radius of the balloon is \( r = 5 \, \text{in} \) when we want to find the rate of change of the radius.

We need to determine \( \frac{dr}{dt} \) when \( r = 5 \).

### Step 1: Relation between Volume and Radius
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]

### Step 2: Differentiate with respect to time
Taking the derivative of both sides with respect to time \( t \):
\[
\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}
\]

### Step 3: Substitute the known values
We know:
- \( \frac{dV}{dt} = -5 \, \text{in}^3/\text{sec} \)
- \( r = 5 \, \text{in} \)

Substituting these values into the differentiated equation:
\[
-5 = 4 \pi (5)^2 \frac{dr}{dt}
\]
\[
-5 = 4 \pi (25) \frac{dr}{dt}
\]
\[
-5 = 100 \pi \frac{dr}{dt}
\]

### Step 4: Solve for \( \frac{dr}{dt} \)
\[
\frac{dr}{dt} = \frac{-5}{100 \pi}
\]
\[
\frac{dr}{dt} = -\frac{1}{20\pi} \, \text{in/sec}
\]

### Conclusion
The radius of the balloon is decreasing at a rate of \( \frac{1}{20\pi} \, \text{in/sec} \).

Would you like more details or have any questions about the solution?

Here are some related questions to deepen your understanding:
1. How does the rate of change of the radius relate to the rate of change of the volume?
2. What would happen to the rate of the radius if the volume decreased faster?
3. How do you interpret a negative rate in the context of this problem?
4. How would the formula change if the balloon were inflating instead of deflating?
5. How would you approach this problem if you were given the surface area instead of the volume?

**Tip:** Always check the units of the rates you are given and the ones you need to find to ensure they match in your final answer.

If a spherical balloon is being deflated at a rate of 5 in^3/sec, at what rate is the radius of the balloon decreasing when r = 5 in?

Explore how to calculate the rate at which the radius of a spherical balloon decreases as it deflates at a rate of 5 in^3/sec, with the radius at 5 inches. This problem covers key calculus concepts such as related rates and geometry of spheres. Suitable for high school calculus students.

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