A chemical manufacturer wants to lease a fleet of 28 railroad tank cars with a combined carrying capacity of 564,000 gallons. Tank cars with three different carrying capacities are available: 6,000 gallons,12,000 gallons, and 24,000 gallons. How many of each type of tank car should be leased?

Let x1 be the number of cars with a 6,000 gallon capacity, x2 be the number of cars with a 12,000 gallon capacity, and x3 be the number of cars with a 24,000 gallon capacity. Select the correct choice below and fill in the answer boxes within your choice.

You forgot to give the choices so I will solve the problem in detail step-by-step explanation and then you can select the correct choice and fill in the answer boxes within your choice correctly.

Given the criteria stated in the problem, we can obtain the following linear system:

[tex]\left \{ {{x_{1} + x_{2} + x_{3} = 28} \atop {6000x_{1} + 12000x_{2} + 24000x_{3} =564000}} \right.[/tex]

The corresponding augmented matrix is

[tex]\left[\begin{array}{ccc|c}1&1&1&28\\6000&12000&24000&564000\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&1&28\\6000&12000&24000&564000\end{array}\right] \frac{1}{6000} R_{1}[/tex]→ [tex]R_{2}\left[\begin{array}{ccc|c}1&1&1&28\\1&2&4&94\end{array}\right] R_{2} +(-1)R_{1}[/tex]

→[tex]R_{2} \left[\begin{array}{ccc|c}1&1&1&28\\0&1&3&66\end{array}\right] R_{1} +(-1)R_{2}[/tex]→[tex]R_{1} \left[\begin{array}{ccc|c}1&0&-2&-38\\0&1&3&66\end{array}\right][/tex]

The linear system corresponding to the reduced form is

[tex]\left \{ {{x_{1} -2x_{3} =-38} \atop {x_{2}+3x_{3} =66}} \right.[/tex]

Since [tex]x_{3}[/tex] is the free variable, we let [tex]x_{3} =t[/tex], where [tex]t[/tex] is any real number. Then the general solution can be written as follows:

[tex]\left \{ \begin{array}{ccc}x_{1} =&-38&+2t\\x_{2} =&66&-3t\\x_{3} =&t\end{array}\right[/tex]

where [tex]t[/tex] is any real number.

However, the definitions of [tex]x_{1}[/tex],[tex]x_{2}[/tex] and [tex]x_{3}[/tex] imply that the solutions must be non-negative integers. Therefore, we need to derive the possible range of values of [tex]t[/tex] such that the general solution make sense for this problem:

[tex]\left \{ \begin{array}{ccc}-38&+2t&\geq 0\\66&-3t&\geq 0\\t&&\geq 0\end{array}\right[/tex]

Solving the inequalities, we finally obtain all the relevant solutions to the linear system:

[tex]\left \{ \begin{array}{ccc}x_{1} =&-38&+2t\\x_{2} =&66&-3t\\x_{3} =&t\end{array}\right[/tex]

where [tex]t[/tex] is any integer such that 19 [tex]\leq[/tex] [tex]t[/tex] [tex]\leq[/tex] 22.

Thus, you just need to apply the above solution to the choices given to you and fill in the answer boxes correctly.