Question

At Brantley's Bakery, a batch of cookies requires 2 cups of flour and 60 minutes to bake. A cake requires 3 cups of flour and 45 minutes to bake. Brantley has budgeted for 30 cups of flour per day and can bake for up to 9 hours per day. Brantley makes the same amount of profit when he sells a batch of cookies as he does when he sells a cake. How many batches of cookies and how many cakes should Brantley make each day to maximize his profit?

Answer 1

To maximize his profit, Brantley should make 4 batches of cookies and 3 cakes each day.

Step-by-step explanation:

Let's denote the number of batches of cookies as C and the number of cakes as \(K\). The profit for each batch of cookies is the same as the profit for each cake.

The constraints are the budgeted flour and baking time:

`\[2C + 3K \leq 30\] (flour constraint)`

`\[60C + 45K \leq 540\] (time constraint, converted to minutes)`

The objective function is to maximize profit:
`\[P = 4C + 3K\]`

To solve this linear programming problem, we can use the simplex method or a graphical method. For brevity, let's use the graphical method.

Plotting the constraints on a graph, the feasible region is the area where the constraints overlap. The corner points of this region are calculated to determine the maximum profit. Solving the system of equations for these corner points gives the maximum profit at
`\(C = 4\) and \(K = 3\).`

Therefore, Brantley should make 4 batches of cookies and 3 cakes each day to maximize his profit.