Question

A chemical manufacturer wants to lease a fleet of 27 railroad tank cars with a combined carrying capacity of 540,000 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
​
=,x
2
​
=, and x
3
​
=. (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
​
=t+(), x
2
​
=t+(), and x
3
​
=t for ≤t≤s (Simplify your answers. Type integers or simplified fractions.) C. There is no solution.

Answer 1

the correct choice is: B. There are multiple possible combinations of how the tank cars should be leased. The combinations are obtained from the equations x1 = t + (), x2 = t + (), and x3 = t for ≤ t ≤ s.

To solve the problem, we can set up a system of linear equations based on the given information:

Let x1 be the number of cars with a 6,000-gallon capacity.

Let x2 be the number of cars with a 12,000-gallon capacity.

Let x3 be the number of cars with a 24,000-gallon capacity

We can write the following equations:

Equation 1: x1 + x2 + x3 = 27 (the total number of tank cars)

Equation 2: 6,000x1 + 12,000x2 + 24,000x3 = 540,000 (the total carrying capacity in gallons)

Now, let's solve this system of equations to find the values of x1, x2, and x3.

From Equation 1, we can express x1 in terms of x2 and x3:

x1 = 27 - x2 - x3

Substituting this into Equation 2, we have:

6,000(27 - x2 - x3) + 12,000x2 + 24,000x3 = 540,000

Simplifying the equation:

162,000 - 6,000x2 - 6,000x3 + 12,000x2 + 24,000x3 = 540,000

Combining like terms:

6,000x2 + 18,000x3 = 378,000

Dividing both sides of the equation by 6,000:

x2 + 3x3 = 63

We can see that this equation does not uniquely determine the values of x2 and x3. There are multiple combinations of x2 and x3 that satisfy the equation.

Therefore, the correct choice is:

B. There are multiple possible combinations of how the tank cars should be leased. The combinations are obtained from the equations x1 = t + (), x2 = t + (), and x3 = t for ≤ t ≤ s.

Since the problem does not specify any constraints or preferences, we cannot determine a unique solution. Different combinations of x2 and x3 within certain ranges will satisfy the given conditions.