A greeting card manufacturer has 500 boxes of a particular card in warehouse I and 270 boxes of the same card in warehouse II. A greeting card shop in San Jose orders 400 boxes of the card, and another shop in Memphis orders 225 boxes. The shipping costs per box to these shops from the two warehouses are shown in the table. How many boxes should be shipped to each city from each warehouse to minimize shipping costs? What is the minimum cost? (Hint: Use $x$, $400-x, y$, and $225-y$ as the variables.) \begin{tabular}{cc|c|c} & \multicolumn{2}{|c}{ DESTINATION } \\ \hline & San Jose & Memphis \\ \hline Warehouse I & $\$ 0.25$ & $\$ 0.22$ \\ & & & \\ & II & $\$ 0.24$ & $\$ 0.19$ \end{tabular}
Solution
Step 1 ：Let the number of boxes shipped from Warehouse I to San Jose be denoted as \(x\), and the number of boxes shipped from Warehouse I to Memphis be denoted as \(y\). Then, the number of boxes shipped from Warehouse II to San Jose will be \(400 - x\), and the number of boxes shipped from Warehouse II to Memphis will be \(225 - y\).
Step 2 ：We have two constraints here: \(x + y ≤ 500\) and \((400 - x) + (225 - y) ≤ 270\).
Step 3 ：The total cost of shipping is given by the sum of the products of the number of boxes and the cost per box for each route, which is \(0.25x + 0.22y + 0.24(400 - x) + 0.19(225 - y)\).
Step 4 ：We want to minimize this cost, so we need to find the minimum of this function subject to the constraints.
Step 5 ：First, let's simplify the cost function: \(Cost = 0.25x + 0.22y + 96 - 0.24x + 42.75 - 0.19y\) which simplifies to \(Cost = 0.01x + 0.03y + 138.75\).
Step 6 ：Now, let's solve the constraints: \(x + y ≤ 500\) and \(625 - x - y ≤ 270\). Solving these, we get \(x ≥ 355\) and \(y ≥ 0\).
Step 7 ：Substituting these values into the cost function, we get: \(Cost = 0.01(355) + 0.03(0) + 138.75 = $141.30\).
Step 8 ：So, to minimize the cost, 355 boxes should be shipped from Warehouse I to San Jose, and no boxes should be shipped from Warehouse I to Memphis. The remaining boxes should be shipped from Warehouse II. The minimum cost is \(\boxed{$141.30}\).
