Let's determine which equation models the relationship between the time [tex]\( t \)[/tex] in minutes and the quarts of water [tex]\( w \)[/tex] left in the tub.
From the problem, we know the following:
- The tub initially has 50 quarts of water.
- The tub empties at a rate of 2.5 quarts per minute.
We can use these pieces of information to form a linear equation.
When [tex]\( t = 0 \)[/tex], the initial amount of water [tex]\( w \)[/tex] is 50 quarts. Therefore, we start with the point (0, 50).
For every minute that passes, the water decreases by 2.5 quarts. This tells us the water [tex]\( w \)[/tex] will be impacted by the rate over time.
A linear relationship can be expressed as:
[tex]\[ w = w_{initial} - (rate \times t) \][/tex]
Substituting the given values:
[tex]\[ w = 50 - 2.5t \][/tex]
This equation [tex]\( w = 50 - 2.5t \)[/tex] correctly models the situation.
To find the solution when [tex]\( t = 30 \)[/tex] minutes, we can substitute [tex]\( t \)[/tex] with 30 into our equation:
[tex]\[ w = 50 - 2.5t \][/tex]
Substituting [tex]\( t = 30 \)[/tex]:
[tex]\[ w = 50 - 2.5 \times 30 \][/tex]
[tex]\[ w = 50 - 75 \][/tex]
[tex]\[ w = -25 \][/tex]
Therefore, when [tex]\( t = 30 \)[/tex] minutes, the amount of water left in the tub is [tex]\( -25 \)[/tex] quarts.
So, the correct equation that models the relationship is:
[tex]\[ w = 50 - 2.5t \][/tex]
The solution when the time is 30 minutes is:
[tex]\[ w = -25 \text{ quarts} \][/tex]
