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Congratulations! You have just been hired as a manager of Todaro’s, a small business that makes frozen pizzas to sell at local markets. The owner gave you the following information: o Preparation and packaging takes 0.2 hours for each box of 12-inch pizzas and 0.25 hours for each box of 16-inch pizzas. o You can pay the staff for no more than 240 hours of labor each week. o The staff must meet the company quota of producing at least 1,000 boxes of pizza per week. o Todaro’s will make a profit of $2 for each box of 12-inch pizzas and $4 for each box of 16-inch pizzas. How many pizzas of each size must be produced in order to maximize the profit? (15 pts.) Let x represent the number of 12-inch pizzas made and let y represent the number of 16-inch pizzas made.
SO that's the info they give you
a) Write an inequality for each constraint. (Hint: There are 4.)
i don't want to be manager!
ha ha me either lol
so I think a is asking for me to basically write an equation for each of the above bullet points right...?
wait, i think we have to see first that we can only have 240 hours of work total yes?
which means .02x+0.25y < 240
because you have to make at least 1000 pies
ok I gotcha
2x+4y = profit
so those are two of the inequalities...
thats cause of the Todaro’s will make a profit of $2 for each box of 12-inch pizzas and $4 for each box of 16-inch pizzas. How many pizzas of each size must be produced in order to maximize the profit right
yes. but i am still trying to find 4 inequalities, because so far i see only two.
oh ya cause that's not an inequality.... hmmmm
no that is what you are trying to maximize subject to the constraints.
oh ok makes sense
i must me missing something. what other constraints are there?
well what I put on here is all that they give me.... unless some how you use the first bullet to relate to the amount of boxes or something?
i am still looking, but i don't see another.
i take it this is one of those 'linear programming' problems. you graph the inequalities and then look at the corners or something like that yes?
me either... maybe I wasn't completely crazy after all lol
Uhm sure I can show you the other parts of it, like the other questions
b. Graph the feasible region that represents the constraints in (a). (Note, you must submit a graph to earn full credit.) c. Locate and label the vertices for the feasible region. d. Write the objective function. e. Find the number of each type of pizza made that will maximize Todaro’s profit.
oh i know
x>0, y>0 lol
you are in quadrant I
right cause they are both positive numbers
so those are my other 2 ineqaulities?
got it. now we have 4 inequalities. i forgot about the last two trivial ones. ok good. so we graph the region. you are in quadrant I when you graph.
right x>0 y>0 x+y>1000 .02x + .25y < 240
the last one could be written as 2x + 25y < 24000 if it makes it easier
Ok got em' now to graph I use .02x + .25y < 240 riht but I have to solve for x and y?
and of course all my inequalities are actually \[\leq\] and \[\geq\]
oh ok good to know
\[x+y\geq 1000\] graph the line which has slope -1 and crosses through (0,1000) and (1000,0) shade to the right and up
i will do that now.
ok give me on sec to graph it please
ok is there a way I could have you check my graph?
let me know when you have graphed it. it should cross the x axis at (1000,0) and the y-axis at (0,1000)
there is my picture of x + y = 1000
ok so I have mine right cool beans
Locate and label the vertices for the feasible region that's the x axis at (1000,0) and the y-axis at (0,1000) right?
now graph the other one. crosses y axis at (0,960) and the x axis at (1200,0)
you have to graph them both. here is my picture of both of them graphed together.
Ok one sec let me graph mine again lol
ok got it.
now shade the regions. should be to the right and up for x + y = 1000 and down to the left of 20x+25y=24000
so it leaves the little part in between them blank right
right, but we are only concerned with the region that is shaded twice, because both of these inequalities must be satisfied
ok and thats the tiny little part
little triangle on bottom right with vertices (1000,), (12000,0) and something in the middle which we have yet to find. this is our main work now is to find where these lines intersect.
i meant (1000,0) and (12000,0) and (a,b) that we have to solve for. so now we want to know where these lines intersect.
ok and that is what Locate and label the vertices for the feasible region means?
i am not sure what method you are to use, but here is an easy one: \[x+y=1000\] solve for x \[y=1000-x\] \[20x+25y=24000\] replace y by 1000-x \[20x+25(1000-x)=24000\] \[20x+25000-25x=24000\] \[-5x+25000=-24000\] \[-5x=-1000\] \[x=200\]
so the vertices are (1000,0), (12000,0) and (200,800) since if x = 200, y = 800
and to find y all you did was plug x back into the original equation right?
yes either one, but easiest to say since x + y = 1000 then 200 + y = 1000 so y = 800
you can almost see it on the graph.
ok ya that's what I meant
Now it asks... Write the objective function?
that was the thing we are trying to maximize.
i have to go back and see. 2x + 4y = P
oh so that's what the objective function is because your objective is to find the profit? Just want to make sure I'm understanding it
and now you have to check the corners of your region to see which gives the biggest. \[p=2\times 200+4\times 800=400+3200=3600\]
yes you are trying to maximize your profit, so you have to see which is biggest.
you have to check x = 200, y = 800 and x = 1000, y = 0 and x = 12000, y = 0
if i remember correctly you have to check the corners of your region
i will be willing to bet without checking that it will turn out to be (200, 800) that gives you the most.
if so you owe me a pizza, or a job. you can manage the shop, i will deliver the pies.
Ok so when it says to write the objective function do I put 2x + 4y = P or 2(200)+4(800)=3600
the objective function is P=2x+4y
alright cause it's ust asking for the formula/equation I got it
but then you have to answer How many pizzas of each size must be produced in order to maximize the profit.
to do this you have to check which values of x and y make P=2x+4y the biggest
ohhhh ok makes perfect sense ha ha want a promotion.... how about manager!?! lol jk thank you thank you thank you now I can use this to help me with my other problems! How do you add as a fan on here?
but you only have to check the corners of your region. you check P=2x+4y for the corners.
don't know. i guess it is a button to click. i
hope the process is clear. graph the lines, graph the region, check the corners. to complete it you also have to check \[P=2\times 1000+4\times 0\] and \[P=2\times 12000 + 4 \times 0\] and \[P=2\times 200 + 4\times 800\] the last one is the biggest
wait.. and x = 12000, y = 0 wouldn't that make it the biggest?
yes but it was a typo. it is \[2\times 1200\]
the x-intercept is (1200,0) not (12000,0) i typed it wrong
oh i gotcha :) and yes I wrote notes down on how we found everything so I think I understand it pretty well know thank you so very much!
welcome. now i go back to work!
lol sorry thanks