Enormous State University's Business School is buying computers. The school has two models from which to choose, the Pomegranate and the iZac. Each Pomegranate comes with 400 GB of memory and 80 TB of disk space; each iZac has 300 GB of memory and 100 TB of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least $48,000 \mathrm{~GB}$ of memory and at least 12,800 TB of disk space. If the Pomegranate and the iZac cost $\$ 3,000$ each, how many of each should the school buy to keep the cost as low as possible? (If an answer does not exist, enter DNE.) Let $x=$ number of Pomegranates, let $y=$ number of iZacs, and let $c=$ cost (in dollars). Determine the objective and the constraints needed to solve the problem. Objective --Select- $c=$ Constraints \[ \begin{array}{c} \geq 48,000 \\ \geq 12,800 \\ x \geq 0, \quad y \geq 0 \end{array} \] Graph the feasible region and determine the coordinates of all corner points. smallest $x$-value $(x, y)=(\square)$ \[ \begin{aligned} (x, y) & =(\square) \\ \text { largest } x \text {-value } \quad(x, y) & =(\square) \end{aligned} \] The minimum value of $c$ occurs at which corner point? \[ (x, y)=(\square) \] State the minimum value of $c$. \[ c= \] How many of each model should the school buy to keep the cost as low as possible? Pomegranate computers iZac computers
Solution
Step 1 ：This is a linear programming problem. We need to minimize the cost function subject to certain constraints. The cost function is \(c = 3000x + 3000y\) where \(x\) is the number of Pomegranate computers and \(y\) is the number of iZac computers.
Step 2 ：The constraints are \(400x + 300y \geq 48000\) (for memory) and \(80x + 100y \geq 12800\) (for disk space). We also have the constraints \(x \geq 0\) and \(y \geq 0\) since we can't buy negative number of computers.
Step 3 ：We can solve this problem using linear programming methods.
Step 4 ：The result from the calculation shows that the school should buy 60 Pomegranate computers and 80 iZac computers. The minimum cost is $420,000.
Step 5 ：Final Answer: The school should buy \(\boxed{60}\) Pomegranate computers and \(\boxed{80}\) iZac computers. The minimum cost is \(\boxed{420,000}\) dollars.
