## 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......

1. anonymous

part 1?

2. 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.

3. anonymous

SO that's the info they give you

4. anonymous

a) Write an inequality for each constraint. (Hint: There are 4.)

5. anonymous

i don't want to be manager!

6. anonymous

ok

7. anonymous

ha ha me either lol

8. anonymous

so I think a is asking for me to basically write an equation for each of the above bullet points right...?

9. anonymous

yes

10. anonymous

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

11. anonymous

which means .02x+0.25y < 240

12. anonymous

ok...

13. anonymous

x+y>1000

14. anonymous

right....

15. anonymous

because you have to make at least 1000 pies

16. anonymous

ok I gotcha

17. anonymous

2x+4y = profit

18. anonymous

so those are two of the inequalities...

19. 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

20. anonymous

yes. but i am still trying to find 4 inequalities, because so far i see only two.

21. anonymous

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

22. anonymous

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

23. anonymous

oh ok makes sense

24. anonymous

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

25. 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?

26. anonymous

i am still looking, but i don't see another.

27. 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?

28. anonymous

me either... maybe I wasn't completely crazy after all lol

29. anonymous

Uhm sure I can show you the other parts of it, like the other questions

30. 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.

31. anonymous

oh i know

32. anonymous

x>0, y>0 lol

33. anonymous

you are in quadrant I

34. anonymous

right cause they are both positive numbers

35. anonymous

so those are my other 2 ineqaulities?

36. 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.

37. anonymous

right x>0 y>0 x+y>1000 .02x + .25y < 240

38. anonymous

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

39. anonymous

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

40. anonymous

and of course all my inequalities are actually $\leq$ and $\geq$

41. anonymous

oh ok good to know

42. 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

43. anonymous

i will do that now.

44. anonymous

ok give me on sec to graph it please

45. anonymous

ok is there a way I could have you check my graph?

46. 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)

47. anonymous

48. anonymous

there is my picture of x + y = 1000

49. anonymous

ok so I have mine right cool beans

50. 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?

51. anonymous

now graph the other one. crosses y axis at (0,960) and the x axis at (1200,0)

52. anonymous

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

53. anonymous

54. anonymous

Ok one sec let me graph mine again lol

55. anonymous

ok got it.

56. 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

57. anonymous

so it leaves the little part in between them blank right

58. anonymous

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

59. anonymous

ok and thats the tiny little part

60. 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.

61. 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.

62. anonymous

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

63. 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 $20x+25(1000-x)=24000$ $20x+25000-25x=24000$ $-5x+25000=-24000$ $-5x=-1000$ $x=200$

64. anonymous

so the vertices are (1000,0), (12000,0) and (200,800) since if x = 200, y = 800

65. anonymous

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

66. anonymous

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

67. anonymous

you can almost see it on the graph.

68. anonymous

ok ya that's what I meant

69. anonymous

Now it asks... Write the objective function?

70. anonymous

that was the thing we are trying to maximize.

71. anonymous

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

72. 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

73. anonymous

and now you have to check the corners of your region to see which gives the biggest. $p=2\times 200+4\times 800=400+3200=3600$

74. anonymous

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

75. anonymous

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

76. anonymous

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

77. anonymous

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

78. anonymous

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

79. anonymous

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

80. anonymous

the objective function is P=2x+4y

81. anonymous

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

82. anonymous

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

83. anonymous

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

84. 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?

85. anonymous

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

86. anonymous

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

87. 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

88. anonymous

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

89. anonymous

yes but it was a typo. it is $2\times 1200$

90. anonymous

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

91. 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!

92. anonymous

welcome. now i go back to work!

93. anonymous

lol sorry thanks