To approach this problem, let’s start by understanding how the volume of water in the tub changes over time.
1. Given Information:
- The tub is initially filled with 50 quarts of water.
- The water empties at a rate of 2.5 quarts per minute.
2. Establish the Relationship:
- Let [tex]\( w \)[/tex] be the amount of water (in quarts) left in the tub.
- Let [tex]\( t \)[/tex] be the time (in minutes) since the tub started to empty.
3. Formulate the Equation:
To model this relationship mathematically, we can express [tex]\( w \)[/tex] as a function of [tex]\( t \)[/tex]:
[tex]\[
w(t) = \text{initial amount} - (\text{emptying rate} \times \text{time})
\][/tex]
Given the initial amount of water is 50 quarts and the rate of emptying is 2.5 quarts per minute, the equation becomes:
[tex]\[
w(t) = 50 - 2.5t
\][/tex]
4. Validation Using Provided Data:
Let’s validate the equation with the values in the table:
- At [tex]\( t = 0 \)[/tex]:
[tex]\[
w(0) = 50 - 2.5(0) = 50 \, (\text{matches the table})
\][/tex]
- At [tex]\( t = 2 \)[/tex]:
[tex]\[
w(2) = 50 - 2.5(2) = 50 - 5 = 45 \, (\text{matches the table})
\][/tex]
- At [tex]\( t = 4 \)[/tex]:
[tex]\[
w(4) = 50 - 2.5(4) = 50 - 10 = 40 \, (\text{matches the table})
\][/tex]
So, the equation [tex]\( w(t) = 50 - 2.5t \)[/tex] correctly models the relationship.
5. Check the Viable Solution for [tex]\( t = 30 \)[/tex] minutes:
Let’s determine the amount of water left in the tub at [tex]\( t = 30 \)[/tex]:
[tex]\[
w(30) = 50 - 2.5 \times 30
\][/tex]
[tex]\[
w(30) = 50 - 75 = -25
\][/tex]
Therefore, after 30 minutes, the calculation gives us [tex]\(-25\)[/tex] quarts of water left in the tub, which is not possible physically (since negative water amount doesn’t make sense). Hence:
6. Conclusion:
- The correct equation is [tex]\( w(t) = 50 - 2.5t \)[/tex].
- There is not a viable solution when [tex]\( t = 30 \)[/tex] minutes because it results in a negative amount of water, which is not possible in real life.
Thus, the answers are:
1. The equation that models the relationship is [tex]\( w(t) = 50 - 2.5t \)[/tex].
2. There is no viable solution when time is 30 minutes.
