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, [tex]$y$[/tex], is a function of the time in minutes, [tex]$x$[/tex], that it has been draining.
What is the range of this function?
A. all real numbers such that [tex]$y \leq 40$[/tex]
B. all real numbers such that [tex]$y \geq 0$[/tex]
C. all real numbers such that [tex]$0 \leq y \leq 40$[/tex]
D. all real numbers such that [tex]$37.75 \leq y \leq 40$[/tex]
Answer (2)
The maximum amount of water in the bathtub is 40 gallons.
The minimum amount of water in the bathtub is 0 gallons.
The amount of water decreases linearly from 40 to 0 gallons.
The range of the function is 0 ≤ y ≤ 40 .
0 ≤ y ≤ 40
Explanation
Understanding the Problem We are given a table that shows the amount of water remaining in Raj's bathtub as a function of time. We want to find the range of this function, which represents all possible values of y (the amount of water remaining).
Identifying the Maximum Value From the table, we see that when x = 0 (initial time), y = 40 gallons. This is the maximum amount of water in the bathtub. The water drains at a rate of 1.5 gallons per minute.
Identifying the Minimum Value Since the water is draining, the amount of water remaining will decrease over time. The minimum amount of water remaining is 0 gallons, which occurs when the bathtub is empty.
Finding When the Bathtub is Empty We can find the time it takes for the bathtub to be empty by setting y = 0 and solving for x . The equation representing the amount of water remaining is y = 40 − 1.5 x . Setting y = 0 , we have 0 = 40 − 1.5 x . Solving for x , we get 1.5 x = 40 , so $x =
1.5 40 = 2 3 40 = 3 80 ≈ 26.67
minutes.
Determining the Range Since the amount of water starts at 40 gallons and decreases to 0 gallons, the range of the function is all real numbers between 0 and 40, inclusive. Therefore, the range is 0 ≤ y ≤ 40 .
Examples
Imagine you're tracking the fuel level in your car as you drive. The range represents all the possible fuel levels you'll see, from a full tank to an empty one. Similarly, in a recipe, the range of oven temperatures you can use might be between 300°F and 450°F. Understanding the range helps you know the limits of what's possible or safe in different situations.
The range of the function representing the amount of water in Raj's bathtub is from 0 to 40 gallons. Therefore, the correct choice is C : 0 ≤ y ≤ 40 . The bathtub starts full at 40 gallons and drains to empty, defining this range.
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