anonymous
  • anonymous
Can someone help me with a five part word problem? I think it is probably easier than I am making it......
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
part 1?
anonymous
  • anonymous
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.
anonymous
  • anonymous
SO that's the info they give you

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anonymous
  • anonymous
a) Write an inequality for each constraint. (Hint: There are 4.)
anonymous
  • anonymous
i don't want to be manager!
anonymous
  • anonymous
ok
anonymous
  • anonymous
ha ha me either lol
anonymous
  • anonymous
so I think a is asking for me to basically write an equation for each of the above bullet points right...?
anonymous
  • anonymous
yes
anonymous
  • anonymous
wait, i think we have to see first that we can only have 240 hours of work total yes?
anonymous
  • anonymous
which means .02x+0.25y < 240
anonymous
  • anonymous
ok...
anonymous
  • anonymous
x+y>1000
anonymous
  • anonymous
right....
anonymous
  • anonymous
because you have to make at least 1000 pies
anonymous
  • anonymous
ok I gotcha
anonymous
  • anonymous
2x+4y = profit
anonymous
  • anonymous
so those are two of the inequalities...
anonymous
  • anonymous
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
anonymous
  • anonymous
yes. but i am still trying to find 4 inequalities, because so far i see only two.
anonymous
  • anonymous
oh ya cause that's not an inequality.... hmmmm
anonymous
  • anonymous
no that is what you are trying to maximize subject to the constraints.
anonymous
  • anonymous
oh ok makes sense
anonymous
  • anonymous
i must me missing something. what other constraints are there?
anonymous
  • anonymous
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?
anonymous
  • anonymous
i am still looking, but i don't see another.
anonymous
  • anonymous
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?
anonymous
  • anonymous
me either... maybe I wasn't completely crazy after all lol
anonymous
  • anonymous
Uhm sure I can show you the other parts of it, like the other questions
anonymous
  • anonymous
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.
anonymous
  • anonymous
oh i know
anonymous
  • anonymous
x>0, y>0 lol
anonymous
  • anonymous
you are in quadrant I
anonymous
  • anonymous
right cause they are both positive numbers
anonymous
  • anonymous
so those are my other 2 ineqaulities?
anonymous
  • anonymous
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.
anonymous
  • anonymous
right x>0 y>0 x+y>1000 .02x + .25y < 240
anonymous
  • anonymous
the last one could be written as 2x + 25y < 24000 if it makes it easier
anonymous
  • anonymous
Ok got em' now to graph I use .02x + .25y < 240 riht but I have to solve for x and y?
anonymous
  • anonymous
and of course all my inequalities are actually \[\leq\] and \[\geq\]
anonymous
  • anonymous
oh ok good to know
anonymous
  • anonymous
\[x+y\geq 1000\] graph the line which has slope -1 and crosses through (0,1000) and (1000,0) shade to the right and up
anonymous
  • anonymous
i will do that now.
anonymous
  • anonymous
ok give me on sec to graph it please
anonymous
  • anonymous
ok is there a way I could have you check my graph?
anonymous
  • anonymous
let me know when you have graphed it. it should cross the x axis at (1000,0) and the y-axis at (0,1000)
anonymous
  • anonymous
1 Attachment
anonymous
  • anonymous
there is my picture of x + y = 1000
anonymous
  • anonymous
ok so I have mine right cool beans
anonymous
  • anonymous
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?
anonymous
  • anonymous
now graph the other one. crosses y axis at (0,960) and the x axis at (1200,0)
anonymous
  • anonymous
you have to graph them both. here is my picture of both of them graphed together.
anonymous
  • anonymous
1 Attachment
anonymous
  • anonymous
Ok one sec let me graph mine again lol
anonymous
  • anonymous
ok got it.
anonymous
  • anonymous
now shade the regions. should be to the right and up for x + y = 1000 and down to the left of 20x+25y=24000
anonymous
  • anonymous
so it leaves the little part in between them blank right
anonymous
  • anonymous
right, but we are only concerned with the region that is shaded twice, because both of these inequalities must be satisfied
anonymous
  • anonymous
ok and thats the tiny little part
anonymous
  • anonymous
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.
anonymous
  • anonymous
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.
anonymous
  • anonymous
ok and that is what Locate and label the vertices for the feasible region means?
anonymous
  • anonymous
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 \[20x+25(1000-x)=24000\] \[20x+25000-25x=24000\] \[-5x+25000=-24000\] \[-5x=-1000\] \[x=200\]
anonymous
  • anonymous
so the vertices are (1000,0), (12000,0) and (200,800) since if x = 200, y = 800
anonymous
  • anonymous
and to find y all you did was plug x back into the original equation right?
anonymous
  • anonymous
yes either one, but easiest to say since x + y = 1000 then 200 + y = 1000 so y = 800
anonymous
  • anonymous
you can almost see it on the graph.
anonymous
  • anonymous
ok ya that's what I meant
anonymous
  • anonymous
Now it asks... Write the objective function?
anonymous
  • anonymous
that was the thing we are trying to maximize.
anonymous
  • anonymous
i have to go back and see. 2x + 4y = P
anonymous
  • anonymous
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
anonymous
  • anonymous
and now you have to check the corners of your region to see which gives the biggest. \[p=2\times 200+4\times 800=400+3200=3600\]
anonymous
  • anonymous
yes you are trying to maximize your profit, so you have to see which is biggest.
anonymous
  • anonymous
you have to check x = 200, y = 800 and x = 1000, y = 0 and x = 12000, y = 0
anonymous
  • anonymous
if i remember correctly you have to check the corners of your region
anonymous
  • anonymous
i will be willing to bet without checking that it will turn out to be (200, 800) that gives you the most.
anonymous
  • anonymous
if so you owe me a pizza, or a job. you can manage the shop, i will deliver the pies.
anonymous
  • anonymous
Ok so when it says to write the objective function do I put 2x + 4y = P or 2(200)+4(800)=3600
anonymous
  • anonymous
the objective function is P=2x+4y
anonymous
  • anonymous
alright cause it's ust asking for the formula/equation I got it
anonymous
  • anonymous
but then you have to answer How many pizzas of each size must be produced in order to maximize the profit.
anonymous
  • anonymous
to do this you have to check which values of x and y make P=2x+4y the biggest
anonymous
  • anonymous
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?
anonymous
  • anonymous
but you only have to check the corners of your region. you check P=2x+4y for the corners.
anonymous
  • anonymous
don't know. i guess it is a button to click. i
anonymous
  • anonymous
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
anonymous
  • anonymous
wait.. and x = 12000, y = 0 wouldn't that make it the biggest?
anonymous
  • anonymous
yes but it was a typo. it is \[2\times 1200\]
anonymous
  • anonymous
the x-intercept is (1200,0) not (12000,0) i typed it wrong
anonymous
  • anonymous
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!
anonymous
  • anonymous
welcome. now i go back to work!
anonymous
  • anonymous
lol sorry thanks

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