In the given context of Arielle filling the kiddie pool with water using two hoses, let's break down the rational expression [tex]\(\frac{2x + 6(x+3)}{2x + 3}\)[/tex].
1. The numerator of the rational expression represents the total amount of water added to the pool (in gallons) over time.
Let's analyze the numerator [tex]\(2x + 6(x+3)\)[/tex]:
- The term [tex]\(2x\)[/tex] likely represents the water added by the first hose over [tex]\(x\)[/tex] minutes.
- The term [tex]\(6(x+3)\)[/tex] represents another amount of water, which is the rate per minute of the second hose over [tex]\(x+3\)[/tex] minutes (i.e., x minutes of the first hose, plus 3 additional minutes).
Therefore, [tex]\(2x + 6(x+3)\)[/tex] captures the total water from both hoses up to the current time.
2. The denominator of the rational expression represents the total time in minutes during which water is entering the pool.
The denominator [tex]\(2x + 3\)[/tex] indicates the combined time for which both hoses have contributed. The term [tex]\(2x\)[/tex] accounts for the first hose (with a rate of 2 gallons per minute) operating for [tex]\(x\)[/tex] minutes, and the second hose operation augmented by the 3 additional minutes.
Therefore, the complete sentences would be:
- The numerator of the rational expression represents the total amount of water added to the pool (in gallons).
- The denominator of the rational expression represents the total time during which water is entering the pool (in minutes).
