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.
Write the system of linear inequalities to represent this situation.
Sketch the feasible region.
Write the optimization equation.
Determine the maximum profit.
Let x be the number of cakes and y be the number of pies.
The time constraint for preparation is:
2x + 3y ≤ 108
The time constraint for decorating is:
x + 0.5y ≤ 20
The non-negativity constraint is:
x ≥ 0
y ≥ 0
The optimization equation is:
Profit = 25x + 12y
To find the feasible region, we need to plot the lines of the inequalities on a graph and shade the region that satisfies all the constraints.
For the first constraint, 2x + 3y ≤ 108, we can rewrite it as y ≤ (108 - 2x)/3.
Plotting this line, we get a straight line with a slope of -2/3 passing through (0, 36) and (54, 0).
For the second constraint, x + 0.5y ≤ 20, we can rewrite it as y ≤ 40 - 2x.
Plotting this line, we get a straight line with a slope of -2 passing through (0, 40) and (20, 0).
The non-negativity constraint x ≥ 0 means the feasible region is to the right of the y-axis.
The non-negativity constraint y ≥ 0 means the feasible region is above the x-axis.
Combining all the constraints, the feasible region is the shaded area that satisfies all the conditions:
```
|
|
40 +-----------------+
| +++ \
| +++ +
| +++ |
36 +-----+-----+-----+-
| / ++ |
| / + |
| + ++ |
24 +---+----+----+----+
| | |
| | |
| + +++ |
12 +---+----+----+----+
| \ ++ |
| \ ++ |
| +++ ++ |
0 +----+----+-------+
0 20 40 60 80
```
The maximum profit can be found at the corner point (0, 36) since this point is the highest in the feasible region.
Substituting x = 0 and y = 36 into the profit equation, we get:
Profit = 25(0) + 12(36) = 432
Therefore, the maximum profit the bakery can make is $432.
To represent the given situation as a system of linear inequalities, we can define the following variables:
Let x be the number of cakes produced per day.
Let y be the number of pies produced per day.
We are given the following constraints:
1. Preparation time constraint:
2 hours of preparation per cake and 3 hours of preparation per pie:
2x + 3y ≤ 108
2. Decorating time constraint:
1 hour of decoration per cake and 0.5 hours of decoration per pie:
x + 0.5y ≤ 20
3. Non-negative constraint:
x, y ≥ 0 (since we cannot produce a negative number of cakes or pies)
The feasible region is the region where all the constraints are satisfied. Let's now sketch the feasible region:
^
|
A____|_____
| /| \
5 | / | \
| / | \
|/____|_________\
0 5 10 15 20 25 30 35 40 45 50 55 60
We can now write the optimization equation:
Maximize the profit: P = 25x + 12y
To determine the maximum profit, we need to find the vertex (corner point) of the feasible region that gives the highest profit. This can be found by evaluating the profit equation at each vertex and finding the maximum value.
By analyzing the feasible region, we can see that the vertices are at points:
A: (0, 0)
B: (0, 20)
C: (27, 6)
D: (54, 0)
Evaluating the profit equation at each vertex:
A: P(0, 0) = 25(0) + 12(0) = $0
B: P(0, 20) = 25(0) + 12(20) = $240
C: P(27, 6) = 25(27) + 12(6) = $717
D: P(54, 0) = 25(54) + 12(0) = $1350
Thus, the maximum profit of $1350 can be achieved by producing and selling 54 cakes and 0 pies per day.