Raj's bathtub is clogged and is draining at a rate of 1.5 gallons of water per minute. The table shows that the amount of water remaining in the bathtub, $y$, is a function of the time in minutes, $x$, that it has been draining.

$x$ $y$

0 40

0.5 39.25

1 38.5

1.5 37.75

What is the range of this function?

Aall real numbers such that $y \leq 40$
Ball real numbers such that $y \geq 0$
Call real numbers such that $0 \leq y \leq 40$
Dall real numbers such that $37.75 \leq y \leq 40$

To determine the range of the function representing the amount of water remaining in the bathtub, we first model the relationship between the time $x$ (in minutes) and the amount of water $y$ (in gallons). From the given data points, we can see that the amount of water decreases linearly. Starting at $x = 0$, the amount of water is $y = 40$. As time increases by $0.5$ minutes, the amount of water decreases by $0.75$ gallons (since $40 - 39.25 = 0.75$). This confirms a constant rate of change:
\[\text{rate} = \frac{-0.75 \text{ gallons}}{0.5 \text{ minutes}} = -1.5 \text{ gallons/minute}.\]
Thus, the function can be expressed as:
\[y = 40 - 1.5x.\]

Identifying Constraints and Boundary Points

The physical context provides two important constraints:

Time must be non-negative, so $x \geq 0$.
The amount of water cannot be negative, so $y \geq 0$.

The maximum amount of water in the bathtub is at the start, when $x = 0$. Substituting this into our function:
\[y = 40 - 1.5(0) = 40.\]
The bathtub becomes empty when the amount of water reaches $0$. We set $y = 0$ and solve for $x$:
\[0 = 40 - 1.5x\]
\[1.5x = 40\]
\[x = \frac{40}{1.5} = \frac{80}{3} \approx 26.67 \text{ minutes}.\]
So, the drainage process spans from $x = 0$ to $x = \frac{80}{3}$.

Determining the Range

The range of a function is the set of all possible output values ($y$). Since the function $y = 40 - 1.5x$ is linear and decreasing, its minimum value in the given interval occurs at the largest value of $x$ (when the tub is empty), which is $y = 0$. Its maximum value occurs at the smallest value of $x$ (at the start), which is $y = 40$. Therefore, the set of all possible amounts of water in the bathtub is all real numbers $y$ such that $0 \leq y \leq 40$. Comparing this to the given options, the correct range is indeed the interval from $0$ to $40$.

Examples

This concept of range in linear functions is widely applied in resource management. For instance, if you have a fuel tank in a car or a water tank in a house, the linear model helps predict how much resource remains at any given time until depletion. By identifying the initial capacity (the maximum value of the range) and the depletion time (where the range reaches zero), you can plan for refueling or maintenance effectively, ensuring you never run out of essential resources.