Question

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

Answer 1

part 1?

Answer 2

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.

Answer 3

SO that's the info they give you

Answer 4

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

Answer 5

i don't want to be manager!

Answer 6

ok

Answer 7

ha ha me either lol

Answer 8

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

Answer 9

yes

Answer 10

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

Answer 11

which means .02x+0.25y < 240

Answer 12

ok...

Answer 13

x+y>1000

Answer 14

right....

Answer 15

because you have to make at least 1000 pies

Answer 16

ok I gotcha

Answer 17

2x+4y = profit

Answer 18

so those are two of the inequalities...

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

oh ok makes sense

Answer 24

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

Answer 25

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?

Answer 26

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

Answer 27

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?

Answer 28

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

Answer 29

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

Answer 30

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.

Answer 31

oh
i know

Answer 32

x>0, y>0 lol

Answer 33

you are in quadrant I

Answer 34

right cause they are both positive numbers

Answer 35

so those are my other 2 ineqaulities?

Answer 36

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.

Answer 37

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

Answer 38

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

Answer 39

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

Answer 40

and of course all my inequalities are actually \[\leq\] and \[\geq\]

Answer 41

oh ok good to know

Answer 42

\[x+y\geq 1000\] graph the line which has slope -1 and crosses through (0,1000) and (1000,0) shade to the right and up

Answer 43

i will do that now.

Answer 44

ok give me on sec to graph it please

Answer 45

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

Answer 46

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

Answer 47

there is my picture of x + y = 1000

Answer 48

ok so I have mine right cool beans

Answer 49

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?

Answer 50

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

Answer 51

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

Answer 52

Ok one sec let me graph mine again lol

Answer 53

ok got it.

Answer 54

now shade the regions. should be to the right and up for x + y = 1000 and down to the left of 20x+25y=24000

Answer 55

so it leaves the little part in between them blank right

Answer 56

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

Answer 57

ok and thats the tiny little part

Answer 58

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.

Answer 59

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.

Answer 60

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

Answer 61

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

Answer 62

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

Answer 63

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

Answer 64

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

Answer 65

you can almost see it on the graph.

Answer 66

ok ya that's what I meant

Answer 67

Now it asks... Write the objective function?

Answer 68

that was the thing we are trying to maximize.

Answer 69

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

Answer 70

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

Answer 71

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

Answer 72

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

Answer 73

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

Answer 74

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

Answer 75

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

Answer 76

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

Answer 77

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

Answer 78

the objective function is P=2x+4y

Answer 79

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

Answer 80

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

Answer 81

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

Answer 82

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?

Answer 83

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

Answer 84

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

Answer 85

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

Answer 86

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

Answer 87

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

Answer 88

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

Answer 89

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!

Answer 90

welcome. now i go back to work!

Answer 91

lol sorry thanks