Question

A company that rents small moving trucks wants to purchase 25 trucks with a combined capacity of 28,000 cubic feet. Three different types of trucks are available: a 10-foot truck with a capacity of 350 cubic feet, a 14-foot truck with a capacity of 700 cubic feet, and a 24-foot truck with a capacity of 1,400 cubic feet. How many of each type of truck should the company purchase

Answer 1

Answer:

We have 4 solutions:

• No 10-foot truck, 10  14-foot trucks, and 15 24-foot trucks
• 2 10-foot trucks, 7 14-foot trucks, and 16 24-foot trucks
• 4 10-foot trucks, 4  14-foot trucks, and 17 24-foot trucks
• 6 10-foot trucks, 1 14-foot trucks, and 18 24-foot trucks

Step-by-step explanation:

Let the number of 10-foot truck with a capacity of 350 cubic feet purchased=a

Let the number of 14-foot truck with a capacity of 700 cubic feet purchased=b

Let the number of 24-foot truck with a capacity of 1,400 cubic feet purchased=c

The company wants to purchase 25 trucks, therefore.

• a+b+c=25

Furthermore, the combined capacity of the trucks is 28,000 cubic feet.

• 350a+700b+1400c=28000

Since the number of equations is less than the number of variables, you can not use a matrix equation to solve this problem.  The solution is most easily found using an augmented matrix.  

The augmented matrix is presented below:  

$\left[\begin{array}{ccc|c}1&1&1&25\\350&700&1400&28000\end{array}\right]$

Using the calculator, the reduced row echelon form is:

$\left[\begin{array}{ccc|c}1&0&-2&-30\\0&1&3&55\end{array}\right]$

where  

a- 2c=-30 means a =2c-30

b+3c=55 means b= 55-3c

We alter the value of c as long as neither a nor b becomes negative. Suitable values for c are 15, 16, 17, and 18:

$\left|\begin{array}{|c||c||c|}a=2c-30&b=55-3c&c\\0&10&15\\2&7&16\\4&4&17\\6&1&18\end{array}\right|$

We can easily  verify that, for each solution, the number of trucks adds up to 25 and the fleet capacity is 28,000 cubic feet.

We therefore have 4 solutions:

• No 10-foot truck, 10  14-foot trucks, and 15 24-foot trucks
• 2 10-foot trucks, 7 14-foot trucks, and 16 24-foot trucks
• 4 10-foot trucks, 4  14-foot trucks, and 17 24-foot trucks
• 6 10-foot trucks, 1 14-foot trucks, and 18 24-foot trucks