First, let's determine the correct equation that models the relationship between the time [tex]\( t \)[/tex] and the quarts of water [tex]\( w \)[/tex].
The tub initially has 50 quarts of water, and it empties at a rate of 2.5 quarts per minute. We can express the amount of water left in the tub, [tex]\( w \)[/tex], as a function of time, [tex]\( t \)[/tex]. The relationship can be modeled as follows:
[tex]\[ w = 50 \, \text{quarts} - (2.5 \, \text{quarts/min} \times t \, \text{min}) \][/tex]
So, the correct equation is:
[tex]\[ w = 50 - 2.5t \][/tex]
From the given options, the correct equation is:
[tex]\[ w = 50 - 2.5t \][/tex]
Next, let's calculate the amount of water left in the tub when the time [tex]\( t \)[/tex] is 30 minutes.
Using the equation [tex]\( w = 50 - 2.5t \)[/tex]:
[tex]\[ w = 50 - 2.5 \times 30 \][/tex]
[tex]\[ w = 50 - 75 \][/tex]
[tex]\[ w = -25 \][/tex]
This would mean that after 30 minutes, the equation indicates there would be -25 quarts of water left, suggesting that the tub not only emptied but that the duration surpassed the time required to fully empty it, leading to a non-physical negative amount of water, indicating an empty tub which means there is no water left. Nonetheless mathematically speaking, since we are investigating the result from the given parameters,
the quarts of water left at t = 30 minutes is -25.
So, the complete solution contains the following points:
1. The correct equation modeling the relationship is [tex]\( w = 50 - 2.5t \)[/tex].
2. According to this equation, when [tex]\( t = 30 \)[/tex] minutes, the quarts of water left, [tex]\( w \)[/tex], is -25 quarts.
In summary:
- The equation: [tex]\( w = 50 - 2.5t \)[/tex]
- The water left when [tex]\( t = 30 \)[/tex] minutes: -25 quarts
