Question 1198757: A company makes two types of biscuits: Jumbo and Regular. The oven can cook at most 400 biscuits per day. Each jumbo biscuit requires 2 oz of flour, each regular biscuit requires 1 oz of flour, and there is 600 oz of flour available. The income from each jumbo biscuit is $0.07 and from each regular biscuit is $0.12 . How many of each size biscuit should be made to maximize income? What is the maximum income?
The company should make how many jumbo and how many regular biscuits.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = number of jumbo biscuits
y = number of regular biscuits
your constraint inequalities are:
x + y <= 400
2x + y <= 600
your objective function is:
income = .07x + .12y
x + y <= 400 says that the total number of biscuits has to be less than or equal to 400.
2x + y <= 600 says that the total amount of flour has to be less than 600 ounces.
there are 2 ounces of flour for each jumbo biscuit and 1 ounce of flour for each regular biscuit.
income = .07x + .12y says that income = 7 cents for each jumbo biscuit and 12 cents for each regular biscuit.
using the desmos.com/calculator, you would graph the opposite of the inequalities.
the area of the graph that is not shaded is your region of feasibility.
the region of feasibility includes the lines of the inequalities that border it.
the corner points of the region of feasibility is where the maximum income will be.
here is what the graph looks like:
the corner points of the region of feasibility are:
(0,400)
(200,200)
(300,0)
you evaluate the objective function at each corner point to find the maximum income.
at (0,400), the income is 0 * .07 + 400 * .12 = 48
at (200,200), the income is 200 * .07 + 200 * .12 = 38
at (300,0), the income is 300 * .07 = 21
the maximum income is at the point (0,400)
that would require no large biscuits and 400 regular biscuits.
Answer by ikleyn(52782)  (Show Source):
You can put this solution on YOUR website! .
A company makes two types of biscuits: Jumbo and Regular.
The oven can cook at most 400 biscuits per day.
Each jumbo biscuit requires 2 oz of flour, each regular biscuit requires 1 oz of flour,
and there is 600 oz of flour available.
The profit from each jumbo biscuit is $0.07 and from each regular biscuit is $0.12 .
How many of each size biscuit should be made to maximize profit ?
What is the maximum profit ?
The company should make how many jumbo and how many regular biscuits.
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In order for all the words and the terms were correct, I REPLACED in your post
each word "income" by "profit", at each appearance.
One possible way to solve the problem is to do it as tutor @Theo did it.
But there is another, much shorter and much more attractive (and more educational) way,
when your brain does work, instead of running formal procedures.
As you read the problem, you see that each Jumbo biscuit requires more flour than
each Regular biscuit, but provides lesser profit than Regular biscuit.
It means that the most aggressive strategy should work, when the company produces
Regular biscuits ONLY and does not produce Jumbo biscuits, at all.
It is so OBVIOUS, that I will not spend my and your time for explanations.
And since we just chose the strategy, we simply need to divide 600 oz by 1 to get
the number of possible Regular biscuits: it is 600/1 = 600.
But at this point, other restriction turns on: the number of biscuits can not
be greater than 400.
+---------------------------------------------------------------+
| Thus the optimal solution/answer is as follows: |
| |
| 400 Regular biscuits should be produced; no Jumbo biscuits; |
| optimal (maximum) profit is 400*0.12 = 48 dollars. |
+---------------------------------------------------------------+
Solved, completed and explained.
You do not need to use heavy artillery to solve this joking entertainment problem.
Common sense PLUS a bit of thinking is enough.
I am more than 100% sure that this presented solution is the expected way to solve the problem.
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