Question

a chemical manufacturer wants to lease a fleet of 26 railroad tank cars with a combined carrying capacity of 516,000 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 (x_1) be the number of cars with a 6,000 - gallon capacity, (x_2) be the number of cars with a 12,000 - gallon capacity, and (x_3) 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. a. the unique solution is (x_1=square), (x_2 = square), and (x_3=square) (simplify your answers.) b. there are multiple possible combinations of how the tank cars should be leased. the combinations are obtained from the equations (x_1=square+square t), (x_2=square+square t), and (x_3 = t) for (squareleq tleqsquare) (simplify your answers. type integers or simplified fractions.) c. there is no solution.

Explanation:

Step1: Set up equations

We have two - equations based on the given information. The total number of tank - cars is 26, so $x_1 + x_2+x_3=26$. The total carrying capacity is 516000 gallons, so $6000x_1 + 12000x_2+24000x_3=516000$. Divide the second equation by 6000 to simplify: $x_1 + 2x_2+4x_3 = 86$.
From the first equation $x_1=26 - x_2 - x_3$. Substitute this into the second simplified equation:
$(26 - x_2 - x_3)+2x_2+4x_3 = 86$.

Step2: Simplify the equation

Combine like - terms: $26+x_2 + 3x_3=86$. Then $x_2=60 - 3x_3$.
Since $x_1,x_2,x_3\geq0$ and are integers, and $x_1 = 26-(60 - 3x_3)-x_3=2x_3 - 34$.
Set up the inequalities: $x_1=2x_3 - 34\geq0$, $x_2=60 - 3x_3\geq0$, and $x_3\geq0$.
From $x_1 = 2x_3 - 34\geq0$, we get $x_3\geq17$. From $x_2=60 - 3x_3\geq0$, we get $x_3\leq20$.
When $x_3 = 17$, $x_2=60-3\times17 = 9$, $x_1=2\times17 - 34 = 0$.
When $x_3 = 18$, $x_2=60 - 3\times18=6$, $x_1=2\times18 - 34 = 2$.
When $x_3 = 19$, $x_2=60 - 3\times19 = 3$, $x_1=2\times19 - 34 = 4$.
When $x_3 = 20$, $x_2=60 - 3\times20 = 0$, $x_1=2\times20 - 34 = 6$.
So there are multiple possible combinations.

Answer:

B. There are multiple possible combinations of how the tank cars should be leased. The combinations are obtained from the equations $x_1 = 2t-34$, $x_2=60 - 3t$, and $x_3=t$ for $17\leq t\leq20$ (where $t$ is an integer)