Question

You are going to make and sell bread. A loaf of Irish soda bread is made with 2 c flour and 1/4 c sugar. Kngelhopf cake is made with 4 c flour and 1 c sugar. You will make a profit of $1.50 on each loaf of Irish Soda Bread and a profit of $4 on each Kugelhopf cake. You have 16 c flour and 3 c sugar.

A.) How many of each kind of bread should you make to maximize the profit?
B.) What is the maximum profit?

Show the formulas please.

Answer 1

A) You should make 4 loaf of Irish soda bread and 2 Kugelhopf cake to maximize the profit.

B) The maximum profit is $14.

Step-by-step explanation

Suppose, you should make
`x` loaf of Irish Soda Bread and
`y` 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
`x` loaf of bread is
`2x` c and the amount of sugar is
`(x)/(4)` c.

Kngelhopf cake is made with 4 c flour and 1 c sugar. So, the amount of flour in
`y` loaf of bread is
`4y` c and the amount of sugar is
`1y` c.

You have total 16 c flour and 3 c sugar. So, the constraints will be.......

`2x+4y\leq 16 ;\\ \\ (x)/(4)+1y\leq 3 ; \\ \\ x\geq 0 ; y\geq 0`

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:
`P= 1.50x+4y`

If we graph the above constraints, then the vertices of the common shaded region will be:
`(0, 0), (8, 0), (4, 2)` and
`(0, 3)` (Please refer to the below image for the graph)

For
`(0, 0)` ,
`P= 1.50(0)+4(0)=0`

For
`(8, 0)` ,
`P= 1.50(8)+4(0)=12`

For
`(4, 2)` ,
`P= 1.50(4)+4(2)=14`

For
`(0, 3)` ,
`P= 1.50(0)+4(3)=12`

So, the profit will be maximum, when
`x= 4` and
`y=2`

Thus, you should make 4 loaf of Irish soda bread and 2 Kugelhopf cake to maximize the profit.

The maximum profit will be $14