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.
To solve this problem, let's define our variables:
Let x be the number of loaves of Irish soda bread
Let y be the number of loaves of banana bread
1. To maximize the profit, we need to find the values of x and y that satisfy the given constraints (the available amount of flour and sugar) and maximize the profit. Our objective function to maximize the profit is:
Profit = (1.5 * x) + (4 * y)
The constraints are:
Flour constraint: 2x + 4y ≤ 16
Sugar constraint: 0.25x + y ≤ 3
2. To solve this, we will use a graphical method:
a. Graph the inequalities: Start by graphing the line 2x + 4y = 16 and shade the region below or on the line to represent the flour constraint. Similarly, graph the line 0.25x + y = 3 and shade the region below or on the line to represent the sugar constraint.
b. Find the feasible region: The feasible region is the shaded area where the constraints overlap.
c. Find the corner points: Identify the vertices (corner points) of the feasible region.
d. Test the objective function at each corner point: Calculate the profit at each corner point using the objective function.
e. Find the maximum profit: Determine which corner point gives the maximum profit.
Let's go step by step:
Step 1: Graph the inequalities
To graph 2x + 4y ≤ 16, we can rewrite it as y ≤ (16 - 2x)/4
To graph 0.25x + y ≤ 3, we can rewrite it as y ≤ 3 - 0.25x
Step 2: Graph the lines
Graph the lines y = (16 - 2x)/4 and y = 3 - 0.25x and shade the regions below or on the lines.
Step 3: Find the feasible region
The feasible region is the shaded area where the regions below or on the lines overlap.
Step 4: Find the corner points
Identify the intersection points between the lines that form the feasible region. These intersection points are the corner points.
Step 5: Test the objective function at each corner point
Plug in the x and y values of each corner point into the objective function (Profit = 1.5x + 4y) to find the profit at those points.
Step 6: Find the maximum profit
Identify the corner point that gives the maximum profit. This point represents the number of each type of bread you should make to maximize your profit.
Step 7: Calculate the maximum profit
Plug in the x and y values of the corner point with the maximum profit into the objective function to find the maximum profit.
I hope this step-by-step explanation helps!
To solve this problem, we need to set up equations based on the given information and then find the optimal solution. Let's break it down step by step:
1. Define Variables:
Let's say x represents the number of loaves of Irish soda bread, and y represents the number of loaves of banana bread.
2. Set up Equations:
We need to make sure we don't exceed the available resources, so we have the following constraints:
- Flour Constraint:
2x + 4y ≤ 16 (since each Irish soda bread requires 2 cups of flour, and each banana bread requires 4 cups of flour, and we have a total of 16 cups available)
- Sugar Constraint:
1/4x + 1y ≤ 3 (since each Irish soda bread requires 1/4 cup of sugar, and each banana bread requires 1 cup of sugar, and we have a total of 3 cups available)
We also need to consider non-negative constraints:
- x ≥ 0 (the number of loaves of Irish soda bread cannot be negative)
- y ≥ 0 (the number of loaves of banana bread cannot be negative)
3. Objective Function:
We want to maximize the profit, so the objective function is:
- Profit: P = 1.5x + 4y
4. Graphical Representation:
We can represent the constraints and the objective function on a graph to visually analyze the feasible region and find the optimal solution.
5. Solve for the Optimal Solution:
By examining the feasible region on the graph, we can find the intersection point(s) of the constraints. There are several methods to find the optimal solution, such as graphical method, simplex method, or using a solver in spreadsheet software.
In this case, we can use the graphical method and identify the point on the graph where the profit is maximized. This point will correspond to the number of loaves of Irish soda bread (x) and banana bread (y) that should be produced.
Once we have the optimal solution, we can substitute the values of x and y into the profit equation P = 1.5x + 4y to find the maximum profit.
I hope this explanation helps you set up the equations and solve the problem!