Let's address each part of the problem:

Part (a)

Yes, this situation represents a function. In this context:

• Independent Variable: Time (in minutes), because it progresses regardless of the water level.
• Dependent Variable: The amount of water left in the bathtub (in gallons), which depends on the elapsed time.

Part (b)

To write an equation, let:

• $V$ represent the volume of water in gallons,
• $t$ represent the time in minutes.

Since the bathtub starts with 20 gallons and drains at a rate of 2.5 gallons per minute, the equation is: $V = 20 - 2.5t$

Part (c)

Domain: The domain is the set of possible values for $t$ (time). Since the bathtub can only drain until it's empty, $t$ ranges from 0 to the time when the bathtub is empty.

To find the time when the bathtub is empty, set $V = 0$: $0 = 20 - 2.5t \Rightarrow t = \frac{20}{2.5} = 8$ Thus, the domain is $0 \leq t \leq 8$.

Discrete or Continuous: The domain is continuous because time is a continuous variable, meaning it can take any value within the interval $[0, 8]$.

Part (d)

Range: The range is the set of possible values for $V$. As $t$ goes from 0 to 8:

• When $t = 0$, $V = 20$ (initial amount).
• When $t = 8$, $V = 0$ (completely drained).

Therefore, the range is $0 \leq V \leq 20$.

Part (e)

To graph the function, plot $V = 20 - 2.5t$ with:

• $t$-axis (horizontal) ranging from 0 to 8,
• $V$-axis (vertical) ranging from 0 to 20.

This line will start at $(0, 20)$ and end at $(8, 0)$, representing the linear decrease of water over time.

Would you like me to create the graph or go into more detail on any part?

Related Questions:

1. What would happen to the equation if the draining rate was changed to 3 gallons per minute?
2. How would the domain and range change if the bathtub started with 25 gallons instead of 20?
3. Could this scenario still be considered a function if water was added intermittently?
4. How does the slope of the graph relate to the draining rate?
5. How would you interpret the intercepts of the graph?

Tip:

Remember, in real-world applications, identifying independent and dependent variables can help understand the relationship and behavior of the situation over time.