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.

Question

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!

Answer 1

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

Answer 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.