To determine the range of the function representing the amount of water remaining in Raj's bathtub, let's break down the problem in a detailed and step-by-step manner.
1. Understand the Initial Condition and the Drain Rate:
- Initial amount of water in the bathtub: 40 gallons.
- Draining rate: 1.5 gallons per minute.
2. The Function of Water Drainage:
- Let [tex]\( y \)[/tex] be the amount of water remaining (in gallons) in the bathtub.
- Let [tex]\( x \)[/tex] be the time in minutes the water has been draining.
- The amount of water remaining can be modeled mathematically as:
[tex]\[
y = 40 - 1.5x
\][/tex]
3. Determining the Maximum and Minimum Values:
- At the beginning (i.e., [tex]\( x = 0 \)[/tex]), the amount of water is at its maximum:
[tex]\[
y_{\text{max}} = 40 - 1.5 \cdot 0 = 40 \text{ gallons}
\][/tex]
- As time progresses, the water continues to drain until it is completely empty. To find when the bathtub is empty, set [tex]\( y = 0 \)[/tex]:
[tex]\[
0 = 40 - 1.5x
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
1.5x = 40
\][/tex]
[tex]\[
x = \frac{40}{1.5} = \frac{80}{3} \approx 26.67 \text{ minutes}
\][/tex]
- By the time [tex]\( x = \frac{80}{3} \approx 26.67 \)[/tex], the water amount [tex]\( y \)[/tex] reaches its minimum, which is 0 gallons.
4. Range of the Function:
- The minimum value of [tex]\( y \)[/tex] is 0 gallons (when the bathtub is completely drained).
- The maximum value of [tex]\( y \)[/tex] is 40 gallons (when draining starts).
5. Conclusion:
- Thus, the range of the function [tex]\( y \)[/tex] is the interval from 0 to 40 gallons. This means that the amount of water [tex]\( y \)[/tex] will always be somewhere between 0 and 40 gallons as the water drains.
Therefore, the correct range of the function representing the amount of water remaining in the bathtub is:
[tex]\[
\text{all real numbers such that } 0 \leq y \leq 40
\][/tex]
This matches the third option provided:
[tex]\[
\text{All real numbers such that } 0 \leq y \leq 40
\][/tex]
