SOLUTION: Hello! I am sorry if this problem is really long. This is a midterm homework problem and we never actually covered this material in depth, so i am a bit worried. The homework is w

Question 1073576: Hello!
I am sorry if this problem is really long. This is a midterm homework problem and we never actually covered this material in depth, so i am a bit worried. The homework is worth 50 points. I attempted part of it but I really need help. There are three sections. The first section I started as well as the second.Thank you for your help !
There are three parts to this assignment.
PART 1
"You and your friend have decided to make custom charm bracelets for a charity crafts fair.
You and your friend can make either Small bracelets that yield a profit of 10 dollars each, or Large bracelets that bring in 15 dollars of profit each. It takes 8 minutes to design Small bracelets but only 6 minutes to design the Large bracelets. It also takes 8 minutes to construct a Small bracelet but does take longer - 12 minutes - to construct a Large bracelet. You will be doing the Design work, but only have just 168 minutes to spend. Your friend will be doing the Construction work, but he has only 240 minutes to spend."
For this part of the assignment, create a table that summarizes the information above. Your table should list both the small and large bracelet, the time needed to design and to construct each, and the maximum amount of time that can be spent on each activity. Use the variable X to represent the Small bracelets and Y to represent the Large bracelets."

For part one I started it but i am unsure if i categorized it correctly. I did it basically like thiS but in box form in a Word document.

RESOURCES- NEEDED FOR A SMALL BRACELET (X)- FOR LARGE(Y)- AVAILABLE
DESIGN - 8MIN - 6 MIN - 168MIN
CONSTRUCT- 8MIN - 12 MIN - 140MIN

PART 2 (20 PTS)
"Using the example in part 1 above and using the same document you used to create your data table, perform the following steps, showing all your work including equations, calculations and graphs:
Identify the variables, the constraints, and write the inequalities that can be derived from the information above
Graph the set of feasible solutions
Identify the coordinates of all four corner points"

For this section I came up with this

Define
X= # of small bracelets
Y= # of large bracelets
Four inequalities
x ≥ 0
y ≥ 0
8x +6y ≤ 168
8x +12y ≤ 240
Test the inequality with (1,1)
8(1) +6(1) ≤ 168 14 ≤ 168
8(1) +12(1) ≤ 240 20 ≤ 240
Corner points
(x, y)= (0,20),(0,0),(21,0),(12,12)

P(x,y) = 10 x + 15y
P (0,0) = 0
P (0,20) = 300
P (21,0) = 210
P (12,12) = 300
Maximum profit is 300$

PART 3
"In this part of the assignment you will use the information gathered in parts 1 and 2 to determine which combination of Small and Large bracelets will yield the greatest profits. Using the same document as in parts 1 and 2 above, perform the following steps:
-Write the Profit equation for this scenario
-Using the Profit equation, calculate the profit of using the sets of values -of X and Y calculated above. Show your results in tabular form
-Identify the maximum profit possible for the scenario stated here

Found 2 solutions by KMST, ikleyn:
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
Good work on parts 1 and 2.
Your graph should look like

The profit equation asked for in part 3 is the
P(x,y)=10x+15y that you included in the part 2 of your solution. You also shows showed the calculations for the pairs (x,y) for the corners of the,feasible region,
as asked in part 3. I just do not know how they expect the table to be organized. Maybe 3 columns with X, Y, and P(X,Y) as headings.
You found the profit to be maximum at points (0,20) and (12,12).
That maximum profit was $300.
Because the maximum happens at two corners of the feasible region,
It happens all along the edge of the feasible region connecting those w points.
That is because all the equations and inequalities are linear,
and two lines either intersect, or are parallel, or are the same line.
The feasible region boundary equation,
8x + 12y = 240 ,
and the profit equation for any possible profit P,
10x + 15y = P,
are parallel lines.
For P=0, the profit line passes through the origin.
Each increase in P moves the line, parallel to its previous graph,
and parallel to 8x + 12y = 240.
Eventually, the line would leave the feasible region.
When that happens, you have the maximal feasible profit.
In general, at that point, either the profit line is intersecting a boundary line at a corner,
or it is coinciding with a boundary line all along an edge of the feasible region.
In this case for any conceivable profit,
the profit line had to be parallel to
8x +12y = 240 , or the same line.
To get that maximum profit, you could make
no small bracelets and 20 large ones, or
3 small bracelets and 18 large ones, or
6 small and 16 large bracelets, or
9 small and 14 large bracelets, or
12 small and 12 large bracelets.
All those (x,y) combinations are points on the line connecting (0,20) and (12,12).
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.
To see more similar solved problems, look into the lesson
    - Solving minimax problems by the Linear Programming method
in this site.

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