A) You should make 4 loaf of Irish soda bread and 2 Kugelhopf cake to maximize the profit.
B) The maximum profit is $14.
Explanation
Suppose, you should make [tex]x[/tex] loaf of Irish Soda Bread and [tex]y[/tex] loaf of Kugelhopf cake.
A loaf of Irish soda bread is made with 2 c flour and 1/4 c sugar. So, the amount of flour in [tex]x[/tex] loaf of bread is [tex]2x[/tex] c and the amount of sugar is [tex]\frac{x}{4}[/tex] c.
Kngelhopf cake is made with 4 c flour and 1 c sugar. So, the amount of flour in [tex]y[/tex] loaf of bread is [tex]4y[/tex] c and the amount of sugar is [tex]1y[/tex] c.
You have total 16 c flour and 3 c sugar. So, the constraints will be.......
[tex]2x+4y\leq 16 ;\\ \\ \frac{x}{4}+1y\leq 3 ; \\ \\ x\geq 0 ; y\geq 0[/tex]
You will make a profit of $1.50 on each loaf of Irish Soda Bread and a profit of $4 on each Kugelhopf cake. So, the profit function will be: [tex]P= 1.50x+4y[/tex]
If we graph the above constraints, then the vertices of the common shaded region will be: [tex](0, 0), (8, 0), (4, 2)[/tex] and [tex](0, 3)[/tex] (Please refer to the below image for the graph)
For [tex](0, 0)[/tex] , [tex]P= 1.50(0)+4(0)=0[/tex]
For [tex](8, 0)[/tex] , [tex]P= 1.50(8)+4(0)=12[/tex]
For [tex](4, 2)[/tex] , [tex]P= 1.50(4)+4(2)=14[/tex]
For [tex](0, 3)[/tex] , [tex]P= 1.50(0)+4(3)=12[/tex]
So, the profit will be maximum, when [tex]x= 4[/tex] and [tex]y=2[/tex]
Thus, you should make 4 loaf of Irish soda bread and 2 Kugelhopf cake to maximize the profit.
The maximum profit will be $14
