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To analyze this problem, we need to interpret the graph of \( V'(t) \), which represents the rate of change of the volume of the balloon over time.

1. **Begin to inflate the balloon:** The child begins to inflate the balloon when \( V'(t) \) becomes positive, which indicates that the volume of the balloon is increasing. According to the graph, this happens at \( t = 3 \).

2. **Finish inflating the balloon:** The child stops inflating the balloon when \( V'(t) \) returns to 0, meaning that the volume is no longer changing (the balloon is fully inflated and the child has stopped adding air). This occurs at \( t = 10 \).

3. **Begin to let the air out:** The child begins to let the air out when \( V'(t) \) becomes negative, which shows that the volume is decreasing. This starts at \( t = 15 \), where \( V'(t) \) drops below zero.

Thus, the answers are:
- A. \( t = 3 \)
- B. \( t = 10 \)
- C. \( t = 15 \)

Would you like more details on any part of the explanation?

Here are some follow-up questions for further learning:
1. How would the graph change if the child inflated the balloon more slowly?
2. What does it mean for the volume when \( V'(t) = 0 \)?
3. If \( V'(t) \) were constant at 2, how would the inflation process differ?
4. What would the graph look like if the child inflated and deflated the balloon repeatedly?
5. How would you calculate the total change in volume of the balloon from the graph of \( V'(t) \)?

**Tip:** The area under the curve of \( V'(t) \) represents the net change in volume \( V(t) \).

A child inflates a balloon, admires it for a while, and then lets the air out. The graph provided shows V'(t), the rate of change of the balloon's volume, as a function of time. At what time does the child: A. Begin to inflate the balloon? t = 3, B. Finish inflating the balloon? t = ?, C. Begin to let the air out? t = ?

This problem involves analyzing the graph of V'(t), which represents the rate of change of the balloon's volume. The solution focuses on key points where inflation begins, stops, and deflation starts. Suitable for students learning differentiation and graph interpretation.

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