A local bakery sells pies and cakes. Each pie takes 20 minutes to prepare and 45 minutes to cook. Each cake takes 30 minutes to cook and a total of 40 minutes to mix and decorate. In a given day, the bakery has 16 hours of employee time available for food preparation and 3 ovens available for 8 hours each. The bakery makes a profit of $5.50 on each pie and $6.00 on each cake.
a. Let x = the number of pies and y = the number of cakes the bakery makes in a given day. Translate the constraints into a system of linear inequalities.
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Max z=5.5x +6y
There are three equations which need to be shown. One for the time to cook, one for the employee time, and one for the profit.
1. 45x + 30y < 1440
The total available time to cook is 1440 min. Total time < 1440
The time to cook each pie is 45 min. Total pie time is 45x
The time to cook each cake is 30 min. Total cake time is 30y
Thus total bake time is 45x + 30y which must be less than 1440; hence the answer.
2. 20x + 40y < 960
Preparation time is limited by employee time, which is 960 min.
Cake preparation time is 40 min, y is the number of cakes, hence total cake time is 40y
Pie preparation time is 20 min, x is the number of pies, hence total pie time is 20x
3. 5.50x + 6.00y = max
The objective function is your profit
You make 5.50 on each pie, so your total profit on pies is 5.50x
You make 6.00 on each cake, so your total profit on cakes is 6.00y