Algebra
posted by Skyelar .
Here is the problem: You are going to make ans sell bread. A loaf of Irish soda bread is made with 2 cups flour and 1/4 cup sugar. Banana bread is made with 4 cups flour and 1 cup of sugar. You will make a profit of $1.50 on each loaf of Irish soda bread a nd a profit of $4 on each banana bread. You have 16 cup flour and 3 cup sugar.
1. How many of each bread should you make to maximize the profit??
2. What is the maximum profit
Would someone mind offering a guided explanation of this? I'm not sure how to set up the equations. Thank you!

This is called "linear programming".
let i = number of Irish
let b = number of banana
then profit = p = 1.5 i + 4 b
now lets plot i on the x axis and b on the y axis. For every value of p there is a line on that graph of form:
b = (1.5/4)i + p/4
b = .375 i + .25 p
NOW, find the feasible region on the graph
You only have 16 flours, so there is a line going from (0,4) (8,0). Call that the flour limit line and draw it on your graph
You only have 3 sugars, so there is a line going from (0,3) to (12 ,0). Call it the sugar limit line and draw it on the graph.
the sugar line hits the flour line where?
flower line b = 4  .5 i
sugar line b = 3  .25 i
solve (you could get this from your graph of course)
0 = 1 .25 i
i = 4
b = 2
NOW, we must test the corners for maximum p
corners are
(0,0)
(0,3)
(4,2)
(8,0)
p(0,0) = 0
p(0,3) = 1.5(0)+4*3 = 12
p(4,2) = 1.5*4 + 4*2 = 14
p(8,0) = 1.5(8) +4(0) = 12
so
max profit = 14 at i = 4 and b = 2 
This is a "linear programming" problem.
Let the number of Banana bread be x
and the number of Irish bread be y
from the flour limitation we have
4x + 2y ≤ 16
2x + y ≤ 8
from the sugar limitation we have
(1/4)x + y ≤ 3
x + 4y ≤ 12
when these two are graphed in the first quadrant of a graph, we get a region bounded by the origin, the x and y intercepts closest to the origin and the intersection of the corresponding equations.
The profit equation would be
P = 4x + 1.5y
the slope of that line is 8/3
The farther this line can move away from the origin (a profit of zero) while still within our region, the larger the profit.
So we can move as far as the intersection of
2x+y = 8 and x+4y = 12
I get y = 16/7 but how can we bake 16/7 loafs of bread?
so let y be the closest whole number or y = 2, then x = 3
the profit would be 3(4) + 2(1.5) = 15
Easy Way:
since both x and y must be whole numbers, there are only 5 possible cases
(0,8), (1,6), (2,4), (3,2), and (4,0)
It would be easy to see that (3,2) produces the largest profit.
Respond to this Question
Similar Questions

Math
Here is the question: "To make a pancake mix, add 1/2 cup of sugar, 3 cups of flour, 1 cup of powdered milk, 1 teaspoon baking powder and 1/2 teaspoon salt. Assuming the volume of the baking powder and salt are negligible, how much … 
math
jeanine used 2 1/2 cups of flour to make muffins and 1 3/4 cups of flour to make biscuits. after bortowing 1/4 cup of flour, she had 2 1/4 cups left to make bread. how much flour did she have to begin with? 
algebra
a bread recipe calls for 2/3 cup of rye flour. you only have 3/4 cup on hand. what percent of your original amount of rye flour will remain after baking the bread? 
Math
Taylor plans to use 2 cups of brown sugar in making 8 loaves of whole wheat bread.If this amount of brown sugar is divided equally into 3 parts, what fraction of a cup will there be for each loaf of bread? 
6th grade math
I'm not sure how to solve this problem using unit rate. Jonathan is baking several boxes of bread. He used 9cups of water to 12 cups of flour. How much water was used per cup of flour? 
math 2
Ben is making a bread that calls for 5 cups of flour. He only has 1/2 cup measuring cup. How many times will Ben need to fill the 1/2 cup measuring cup to get the 5 cups of flour? 
Math
Ben is making a bread that calls for 5 cups of flour.He only has a cup 1/2cup measuring cup.how many times will Ben need to fill the 1/2cup measuring cup to get the 5 cups of flour? 
Math
Amy uses 8 cups of flour to make 3 identical loaves of bread.How much flour is in each loaf? 
Math
Jeanine used 21/2 cups of flour to make muffins and 1 3/4 cups of flour to make biscuits. After borrowing 1/4 cup of flour,she had 2 1/4 cups left to make bread. How much flour did she have to begin with? 
Math
Onequarter of a bread recipe calls for 2/3 cup of bread flour. How many cups are needed per recipe?