SOLUTION: How would I set up and solve this equation? Please show all your work 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. K

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: How would I set up and solve this equation? Please show all your work 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. K      Log On


   

Question 902131: How would I set up and solve this equation? Please show all your work
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. Kugelhopf 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 $4.00 of of Kugelhopf. You have 16 c flour and 3 c sugar. How many of each kind of bread should you make to maximize the profit? What is the maximum profit?

So far I know P=1.50x + 4y
.25x+1y≤3
2x+4y≤16
I dont know what to do after please help
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
x = number of loafs of irish soda bread.
y = number of loafs of kugelhopf cake.

a loaf of irish soda bread takes 2 cups flour and 1/4 cup sugar.
a loaf of kugelhopf cake takes 4 cups flour and 1 cup sugar.

profit on irish soda bread is 1.50 per loaf.
profit on kugelhopf is 4.00 per loaf.

you have 16 cups of flour available.
you have 3 cups of sugar available.

your profit equation is:

profit = 1.50*x + 4.00*y

you got that right.

your constraints for flour are:

2x + 4y <= 16

your constraints for sugar are:

(1/4)*x + y <= 3

you got the constraints right.

what's left is to graph the constraints and find the area of feasibility and then find the corners of the area of feasibility and then analyze the profit equation at each of the corners to find the maximum profit.

to graph the equations, you would need to solve for y in each of them.

2x + 4y <= 16 becomes y <= (-2x + 16)/4 which becomes y <= -(1/2)x + 4

(1/4)x + y <= 3 becomes y <= -(1/4)x + 3

an implied constraint is that x >= 0 and y >= 0

your feasible region will be the area on the graph bounded by:

y <= -(1/4)x + 3
y <= -(1/2)x + 4
x >= 0
y >= 0

the graph will look like this:

$$$


the intersection points bounding the area of feasibility are shown on the graph.

you need to evaluate your objective function at each of these intersection points to find the maximum profit.

for example:

at the point (4,2), your profit will be 1.5*4 + 4*2 = 14 dollars.

do the same at all the intersection points and the one with the largest value is the maximum profit.