A local bakery sells cakes and pies. The cakes take 2 hours of preparation time and 1 hour to decorate. The pies take 3 hours of preparation time and 0.5 hours to decorate. The profit is $25 per cake and $12 per pie. The bakery can spend at most 108 hours on preparation and 20 hours on decorating their items per day.
a.Write the system of linear inequalities to represent this situation.
b.Sketch the feasible region.
c.Write the optimization equation.
d.Determine the maximum profit.
Let's represent the number of cakes sold per day as x and the number of pies sold per day as y.
a. The system of linear inequalities is:
2x + 3y ≤ 108 (preparation time constraint)
x + 0.5y ≤ 20 (decorating time constraint)
x ≥ 0 (non-negativity constraint)
y ≥ 0 (non-negativity constraint)
b. To sketch the feasible region, we need to plot the lines representing the equalities in the time constraints and shade the region that satisfies the inequalities. The lines are:
2x + 3y = 108 (preparation time constraint)
x + 0.5y = 20 (decorating time constraint)
The feasible region is the region below or on these lines.
c. The optimization equation is:
Profit = 25x + 12y
d. To determine the maximum profit, we need to find the coordinates of the vertices of the feasible region and evaluate the profit equation at each vertex. The maximum profit will be the highest value obtained.
Analyzing the intersection of the two lines:
2x + 3y = 108
x + 0.5y = 20
By solving this system of linear equations,
x = 12
y = 24
Evaluating the profit equation at this point:
Profit = 25(12) + 12(24)
Profit = 300 + 288
Profit = $588
Therefore, the maximum profit is $588 when the bakery sells 12 cakes and 24 pies.
a. Let's define our variables:
Let x = number of cakes
Let y = number of pies
The system of linear inequalities can be written as follows:
2x + 3y ≤ 108 (preparation time constraint)
x + 0.5y ≤ 20 (decorating time constraint)
x ≥ 0 (non-negativity constraint for cakes)
y ≥ 0 (non-negativity constraint for pies)
b. To sketch the feasible region, we can plot the boundary lines for the inequalities and shade the region that satisfies all the constraints.
Boundary Lines:
2x + 3y = 108 (prep time constraint)
x + 0.5y = 20 (decorating time constraint)
x = 0 (non-negativity constraint for cakes)
y = 0 (non-negativity constraint for pies)
c. The optimization equation is the objective function. In this case, we want to maximize the profit. The profit per cake is $25 and the profit per pie is $12, so the optimization equation is:
Z = 25x + 12y
d. To determine the maximum profit, we need to find the maximum value of Z (profit) within the feasible region. This can be done by evaluating Z at the vertices or corner points of the feasible region. The vertex with the highest Z value will give us the maximum profit.