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anonymous

  • 5 years ago

Can someone help me with a five part word problem? I think it is probably easier than I am making it......

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  1. anonymous
    • 5 years ago
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    part 1?

  2. anonymous
    • 5 years ago
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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.

  3. anonymous
    • 5 years ago
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    SO that's the info they give you

  4. anonymous
    • 5 years ago
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    a) Write an inequality for each constraint. (Hint: There are 4.)

  5. anonymous
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    i don't want to be manager!

  6. anonymous
    • 5 years ago
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    ok

  7. anonymous
    • 5 years ago
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    ha ha me either lol

  8. anonymous
    • 5 years ago
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    so I think a is asking for me to basically write an equation for each of the above bullet points right...?

  9. anonymous
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    yes

  10. anonymous
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    wait, i think we have to see first that we can only have 240 hours of work total yes?

  11. anonymous
    • 5 years ago
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    which means .02x+0.25y < 240

  12. anonymous
    • 5 years ago
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    ok...

  13. anonymous
    • 5 years ago
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    x+y>1000

  14. anonymous
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    right....

  15. anonymous
    • 5 years ago
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    because you have to make at least 1000 pies

  16. anonymous
    • 5 years ago
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    ok I gotcha

  17. anonymous
    • 5 years ago
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    2x+4y = profit

  18. anonymous
    • 5 years ago
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    so those are two of the inequalities...

  19. anonymous
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    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

  20. anonymous
    • 5 years ago
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    yes. but i am still trying to find 4 inequalities, because so far i see only two.

  21. anonymous
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    oh ya cause that's not an inequality.... hmmmm

  22. anonymous
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    no that is what you are trying to maximize subject to the constraints.

  23. anonymous
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    oh ok makes sense

  24. anonymous
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    i must me missing something. what other constraints are there?

  25. anonymous
    • 5 years ago
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    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?

  26. anonymous
    • 5 years ago
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    i am still looking, but i don't see another.

  27. anonymous
    • 5 years ago
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    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?

  28. anonymous
    • 5 years ago
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    me either... maybe I wasn't completely crazy after all lol

  29. anonymous
    • 5 years ago
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    Uhm sure I can show you the other parts of it, like the other questions

  30. anonymous
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    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.

  31. anonymous
    • 5 years ago
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    oh i know

  32. anonymous
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    x>0, y>0 lol

  33. anonymous
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    you are in quadrant I

  34. anonymous
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    right cause they are both positive numbers

  35. anonymous
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    so those are my other 2 ineqaulities?

  36. anonymous
    • 5 years ago
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    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.

  37. anonymous
    • 5 years ago
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    right x>0 y>0 x+y>1000 .02x + .25y < 240

  38. anonymous
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    the last one could be written as 2x + 25y < 24000 if it makes it easier

  39. anonymous
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    Ok got em' now to graph I use .02x + .25y < 240 riht but I have to solve for x and y?

  40. anonymous
    • 5 years ago
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    and of course all my inequalities are actually \[\leq\] and \[\geq\]

  41. anonymous
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    oh ok good to know

  42. anonymous
    • 5 years ago
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    \[x+y\geq 1000\] graph the line which has slope -1 and crosses through (0,1000) and (1000,0) shade to the right and up

  43. anonymous
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    i will do that now.

  44. anonymous
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    ok give me on sec to graph it please

  45. anonymous
    • 5 years ago
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    ok is there a way I could have you check my graph?

  46. anonymous
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    let me know when you have graphed it. it should cross the x axis at (1000,0) and the y-axis at (0,1000)

  47. anonymous
    • 5 years ago
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  48. anonymous
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    there is my picture of x + y = 1000

  49. anonymous
    • 5 years ago
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    ok so I have mine right cool beans

  50. anonymous
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    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?

  51. anonymous
    • 5 years ago
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    now graph the other one. crosses y axis at (0,960) and the x axis at (1200,0)

  52. anonymous
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    you have to graph them both. here is my picture of both of them graphed together.

  53. anonymous
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  54. anonymous
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    Ok one sec let me graph mine again lol

  55. anonymous
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    ok got it.

  56. anonymous
    • 5 years ago
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    now shade the regions. should be to the right and up for x + y = 1000 and down to the left of 20x+25y=24000

  57. anonymous
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    so it leaves the little part in between them blank right

  58. anonymous
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    right, but we are only concerned with the region that is shaded twice, because both of these inequalities must be satisfied

  59. anonymous
    • 5 years ago
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    ok and thats the tiny little part

  60. anonymous
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    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.

  61. anonymous
    • 5 years ago
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    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.

  62. anonymous
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    ok and that is what Locate and label the vertices for the feasible region means?

  63. anonymous
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    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 \[20x+25(1000-x)=24000\] \[20x+25000-25x=24000\] \[-5x+25000=-24000\] \[-5x=-1000\] \[x=200\]

  64. anonymous
    • 5 years ago
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    so the vertices are (1000,0), (12000,0) and (200,800) since if x = 200, y = 800

  65. anonymous
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    and to find y all you did was plug x back into the original equation right?

  66. anonymous
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    yes either one, but easiest to say since x + y = 1000 then 200 + y = 1000 so y = 800

  67. anonymous
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    you can almost see it on the graph.

  68. anonymous
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    ok ya that's what I meant

  69. anonymous
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    Now it asks... Write the objective function?

  70. anonymous
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    that was the thing we are trying to maximize.

  71. anonymous
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    i have to go back and see. 2x + 4y = P

  72. anonymous
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    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

  73. anonymous
    • 5 years ago
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    and now you have to check the corners of your region to see which gives the biggest. \[p=2\times 200+4\times 800=400+3200=3600\]

  74. anonymous
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    yes you are trying to maximize your profit, so you have to see which is biggest.

  75. anonymous
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    you have to check x = 200, y = 800 and x = 1000, y = 0 and x = 12000, y = 0

  76. anonymous
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    if i remember correctly you have to check the corners of your region

  77. anonymous
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    i will be willing to bet without checking that it will turn out to be (200, 800) that gives you the most.

  78. anonymous
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    if so you owe me a pizza, or a job. you can manage the shop, i will deliver the pies.

  79. anonymous
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    Ok so when it says to write the objective function do I put 2x + 4y = P or 2(200)+4(800)=3600

  80. anonymous
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    the objective function is P=2x+4y

  81. anonymous
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    alright cause it's ust asking for the formula/equation I got it

  82. anonymous
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    but then you have to answer How many pizzas of each size must be produced in order to maximize the profit.

  83. anonymous
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    to do this you have to check which values of x and y make P=2x+4y the biggest

  84. anonymous
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    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?

  85. anonymous
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    but you only have to check the corners of your region. you check P=2x+4y for the corners.

  86. anonymous
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    don't know. i guess it is a button to click. i

  87. anonymous
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    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

  88. anonymous
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    wait.. and x = 12000, y = 0 wouldn't that make it the biggest?

  89. anonymous
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    yes but it was a typo. it is \[2\times 1200\]

  90. anonymous
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    the x-intercept is (1200,0) not (12000,0) i typed it wrong

  91. anonymous
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    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!

  92. anonymous
    • 5 years ago
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    welcome. now i go back to work!

  93. anonymous
    • 5 years ago
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    lol sorry thanks

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