Lisa is making cookies to sell at the Annual Dirt Bike Competition. A dozen oatmeal cookies require 3 cups of flour and 2 eggs. A dozen sugar cookies require 4 cups of flour and egg. She has 40 cups of and 20 eggs. She can make no more than dozen oatmeal cookies and no more than 7 dozen sugar cookies, and she earns $3 each dozen oatmeal cookies and 2 for each dozen sugar cookies. How many dozens of each type of cookie should she make to maximize her profit?

Oatmeal cookies = 1 dozen

Sugar cookies = 7 dozen

Explanation:

Objective function:

Maximize:        P = 3x + 2y

where

                       x =  oatmeal cookies

                       y = sugar cookies

Constraints:

Since it takes 2 and 4 cups for oatmeal and sugar cookies respectively, and she has total 40 cups of sugar:

1)     3x+4y ≤ 40

The same for eggs:

2)     2x+y ≤ 20

She can make no more than 1 dozen oatmeal and 7 dozen sugar cookies:

3)      x ≤ 1

4)      y ≤ 7

Now graph these above four constraints equation( see the image below)

Now the corner points are :

(0,7), (1,7), (1,0), (0,0)

Put these corner points in objective function:

for (0,7) :  P = ( 3×0 )  + ( 2×7 ) = 14

for (1,7) :   P = ( 3×1 )  + ( 2×7 ) = 17

for (1,0) :   P = ( 3×1 )  + ( 2×0 ) = 3

for (0,0) :  P = ( 3×0 )  + ( 2×0 ) = 0

Since the maximum profit = 17 comes from the point (1,7), she should make 1 dozen of oatmeal cookies and 7 dozen of sugar cookies.

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