Chapter 7, Problem 75RE

Solution

To calculate: The number of vehicles of each type are needed to deliver all of these.

The numbers are 2 minivans, 4 vans, and 2 trucks

Given Information:

Hix's Discount Electronics has three sizes of high-definition televisions (HDTVs) on sale: small (19-in.), medium (32-in.), and large (47-in.). Hix's has three types of vehicles that they use for deliveries: minivans, vans, and trucks. The minivans can carry 8 small, 3 medium, and 2 large HDTVs; the vans, 15 small, 10 medium, and 6 large; the trucks, 22 small, 20 medium, and 5 large. On the last day of the sale, Hix's has 115 small, 85 medium, and 35 large HDTVs to deliver.

Calculation:

Consider the given information,

Suppose that x is representing number of minivans, and y is number of vans z is representing number of trucks.?

Set up the equations. Considering the small HDTVs, there are 115 on the last day of sale:

  $8x+15y+22z=115$

Considering the medium HDTVs, there are 85 on the last day of sale:

  $3x+10y+20z=85$

Considering the large HDTVs, there are 35 on the last day of sale:

  $2x+6y+5z=35$

Write the system of matrix $AX=B$ .

  $A=\left[\begin{array}{ccc}8& 15& 22\\ 3& 10& 20\\ 2& 6& 5\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],\text{ and }B=\left[\begin{array}{c}115\\ 85\\ 35\end{array}\right]$

And,

  $\begin{array}{c}X=A^{-1}B\\ ={\left[\begin{array}{ccc}8& 15& 22\\ 3& 10& 20\\ 2& 6& 5\end{array}\right]}^{-1}\left[\begin{array}{c}115\\ 85\\ 35\end{array}\right]\left(\text{use calculator for inverse}\right)\\ =\left[\begin{array}{c}1.76856\\ 3.2964\\ 2.33625\end{array}\right]\\ \approx \left[\begin{array}{c}2\\ 4\\ 2\end{array}\right]\end{array}$

Therefore, the required numbers are 2 minivans, 4 vans, and 2 trucks.