1. 3.3 Linear Programming

2. What if we wanted to know how to maximize profits or minimize costs?
We could use an application of linear inequality systems.

3. Ex. Jenny’s Bakery makes two types of birthday cakes: yellow cakes which sell for $25 and strawberry cakes which sell for $35. Both cakes are the same size, but the decorating and assembly time required for the yellow cake is 2 hours, while the strawberry cake takes 3 hours. There are 450 hours of labor available for production. How many of each type should be made to maximize revenue?

4. So to solve this problem, we could use linear programming.
So we will let x = yellow cakes
y = strawberry cakes
The optimization function will be:
f(x,y) = 25x + 35y

5. The linear inequality system would be:
x ≥0
y ≥0
2x + 3y ≤450

6. (0,150)
Feasibility region
100
(0,0)
(225,0)
100

7. So after we graph the linear inequalities, also known as the constraints, we shade the overlapping region known as the feasibility region. The vertices of this region will be plugged into the optimization function to determine maximum revenue.

8. So again, the optimization function was f(x,y) = 25x + 35y
So the maximized revenue would occur if 25 yellow cakes and no strawberry cakes are made.
f(x,y)
( 0, 150)
( 225,0)
( 0,0)
25x + 35y
25(0) + 35(150) = 5250
25(225) +
35(0)=
5625
Maximum
25(0)+