The problem involves a bathtub initially filled with 40 gallons of water, which drains at a rate of 3 gallons per minute. We are asked to find the rate of change and the initial value of the linear function that models the amount of water in the bathtub after it starts draining.

Analysis:

• Initial Value (y-intercept): The bathtub starts with 40 gallons of water, so the initial value is 40 gallons.

• Rate of Change (Slope): Since the water is draining, the rate of change will be negative. The problem states that the water drains at an average of 3 gallons per minute, so the rate of change is $-3$ gallons per minute.

Conclusion:

• The initial value is 40 gallons.
• The rate of change is $-3$ gallons per minute.

Thus, the correct answer is C: The rate of change is $-3$ gallons per minute, and the initial value is 40 gallons.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the rate of change differ if the water was filling the bathtub instead of draining?
2. What is the linear equation that models this situation?
3. How long will it take to completely drain the bathtub?
4. What would the graph of this linear function look like?
5. What does the slope of the line represent in real-life terms?

Tip: In any problem involving rate of change, pay attention to whether the context involves increase (positive slope) or decrease (negative slope).