SOLUTION: Please I appreciate your help with question, just need to check if my working out is correct and try to see how the problem is solved Question B. Maria decides she can produce

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Question 1115931: Please I appreciate your help with question, just need to check if my working out is correct and try to see how the problem is solved
Question
B. Maria decides she can produce rose frames to supplement her income. She calculates that each medium frame requires a 6m length of timber and can be built in four minutes, while a large frame requires a 12m length of timber and can be built in six minutes.
She has a regular supply of suitable timber, a total length of 900 meters per week. She also has a maximum of 8 hours a week to build them.
Maria contacts the manager of a local chain of hardware stores. The manager is willing to stock Maria’s frame if she can guarantee at least 20 of each size per week, but will take no more than 60 of each size per week.
The deal will give Maria a $6 profit on each medium frame produced and a $10 profit on each large frame.
How many of each size frame should Maria produce to maximize her profit?
(Write down the objective function and the constraints and then solve.)
Below is my working out but haven't finished solved the problem
Objective Function
P = $6.00x + $10.00y

Write the constraints
Length constraints: 6x + 12y ≤ 900
Time constraints 4x + 6y ≤ 480
Non-negative constraints x ≥ 0, y ≥ 0
6x + 12y <= 900
4x + 6y <= 480
x ≥ 0, y ≥ 0

Thank you

Found 3 solutions by stanbon, greenestamps, ikleyn:
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
P = $6.00x + $10.00y
Write the constraints
Length constraints: 6x + 12y ≤ 900
Time constraints 4x + 6y ≤ 480
Non-negative constraints x ≥ 0, y ≥ 0
6x + 12y ≤ 900
4x + 6y ≤ 480
x ≥ 0, y ≥ 0
-----
Solve the inequalities for "y" ; Graph each in the first Quadrant;
Find the corners of the inclosed polygon.
Test each corner coordinate pair in the objective function
to find the answer.
-------------
Cheers,
Stan H.
---------------
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


There are several constraints missing from your list:

hardware store constraints: x >= 20; x <= 60; y >= 20; y <= 60

Fortunately, those additional constraints do not change the answer to the problem.

You need to graph the boundary lines for the constraint functions in the first quadrant and find where those boundaries intersect, to determine the feasibility region.

Then evaluate the objective function at each corner of the feasibility region to find where the maximum profit is obtained.
Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.
Comparing with your starting post
    - https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1115623.html
you made a huge progress.   My congratulations !

But you incorrectly wrote these constraints x >= 0, y >= 0. 

Actually, they are  20 <= x <= 60  and  20 <= y <= 60  and  (despite to @greenestamps statement),

these constraints  make influence  to the feasibility domain: this domain is bounded (is entirely inside) 

the rectangle 20 <= x <= 60  and  20 <= y <= 60  shown in the figure below:







So (again), your feasibility domain is the 5-sided (=5-vertices) polygon concluded inside the rectangle and located below the two sloped lines (red and green).


Thus you must calculate and compare the profit function in all 5 vertices of the polygon.

Good luck !


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