Question

Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea
and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A
breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternon blend, how many pounds of each blend should she make to maximi profit?maximum proft is?

Answer 1

a) Representation:
Let X = # pounds in A
Let Y = # pounds in B
Let P = Maximum profit.

b) Set up a objective function:
P = 1.50x + 2.00y

c)Set up constraint inequality:
I don't know this part. It's confusing, please help me, i only need hep this part.

d)Graph inequality and find vertices:
I know this part.

I was looking for answers around the web, i know the correct answer to this problem is 75 packages of the breakfast blend, 40 packages of the afternoon blend; maximum profit $192.50

Answer 2

@Squirrels

Answer 3

@mathmale Can you help out?

Answer 4

Grade A = 45 lbs
Grade B = 70 lbs
Total weight = A+B = 45+70 = 115 pounds of tea in to 115 packages.

Answer 5

Let call "m" = breakfast blend. 1/3A + 2/3B
Let call "n" = afternoon blend. 1/2A + 1/2B

Answer 6

x≥0, y≥ 0
We know the variables must be non negative.

Answer 7

I feel like I'm missing 2 main constraint (restriction) about the breakfast and afternoon blend.

Answer 8

@cram

Answer 9

@dan815

Answer 10

its all corect so far cant help other than that

Answer 11

okay what language do you want to write this program in?

Answer 12

This is a math problem, it's precalculus, it's a method to find the minimize the cost and maximize profit of something. It's not a computer program =)

Answer 13

@M.D.Saxon

Answer 14

\[x + y \le 115 \]
Because we know that what whatever we do, the amount can't exceed over 115 pounds.

Answer 15

Hm, yes, and the "m" and "n" are actually "x" and "y"

x= 1/3A+ 2/3B
y= 1/2A+ 1/2B

so you can plug in x+y <_115 if you need another inequality.

Answer 16

Thanks study100. In the mean time, I'm watching examples from youtube. I hope you guys continue to help me. =)

Answer 17

@phi

Answer 18

@AravindG

Answer 19

https://answers.yahoo.com/question/index?qid=20120810185251AA9Awkn

Answer 20

Yah, i have seen this one before, but it's not how we suppose to solve it. In linear programing problems, we need to follow the following the procedures:

1Choose the unknowns.

2Write the objective function.

3Write the constraints as a system of inequalities.

4 Find the set of feasible solutions that graphically represent the constraints.

5 Calculate the coordinates of the vertices from the compound of feasible solutions.

6 Calculate the value of the objective function at each of the vertices to determine which of them has the maximum or minimum values. It must be taken into account the possible non-existence of a solution if the compound is not bounded.

Answer 21

I was looking for answers around the web, i know the correct answer to this problem is :

75 packages of the breakfast blend, 40 packages of the afternoon blend; maximum profit $192.50

Answer 22

strange thing is that sum of max pounds on that yahoo answer exceeds the original sum, 115 pounds.

their answer add up 212 pounds. Where does all those extra pounds come from?

Answer 23

Corner (x, y) Objective Function "z = 86x + 130y"

(x, y) z = 86(0) + 130(0)
(0, 100) z = 86(0) + 130(100) = $ 13086
(85.56, 0) z = 86(85.56) + 130(0) = $ 7358.16
(0, 110) z = 86(0) + 130(110) = $ 14300
(35, 65) z = 86(35) + 130(65) = $ 11460
(100, 0) z = 86(100) + 130(0) = $ 8600 <----- Minimum cost.

This is an example of another linear programing problem, after we set up the inequality, we will begins to find vertices and substitute x and y axis variables. It is easier to solve rather than yahoo's method.

Answer 24

Do you think we should do like this?

(1/3x + 1/2x) + (2/3y + 1/2y) ≤115

Answer 25

define
x # of breakfast type
y = # of aft type

the amount of type A tea will be
x * ⅓ + y * ½ pounds ≤ 45

the amount of B tea will be
2x/3 + y/2 ≤ 70

these are your two constraints

Answer 26

Let y = # pounds in A
Let y = # pounds in B
Let P = Maximize profit.

P = 1.50x + 2.00y

1/3x + 1/2x ≤ 45
2x/3 + y/2 ≤ 70
x+y ≤115
x≥0
y≥ 0

I think this it is, thanks Phi.

Answer 27

Well, i add up the fractions, i get the following:

5/6x ≤ 45
7/6y ≤ 70
I'm using https://www.desmos.com/calculator to graph these two inequalities. But, it looks odd...

Answer 28

Here is what it looks like,

Answer 29

@phi can you explain how you found the constraints?

Answer 30

@phi May i ask you what variables did you plug into the calculator to get that graph? Would you please screenshot your entire screen please?

Answer 31

x * ⅓ + y * ½ pounds ≤ 45

he simplify it to
1/3y <_ 45 -(1/2)x
y<_ 135 - 3/2x

Same goes with the other equation.

Answer 32

those are the relations you posted above. I used geogebra (free to download)
here with the lines labeled

Answer 33

I think there is some error to our constraints, because:
1/3x + 1/2x ≤ 45 --> total Type A
2/3y + 1/2y ≤ 70 ---> total type B

But, Phi did:
x * ⅓ + y * ½ ≤ 45

the amount of B tea will be
2x/3 + y/2 ≤ 70

Answer 34

@study100
good question. The first thing I had to do was not get confused about the objective function (how to calculate the profit) which has nothing to do with the "feasible region"

we are left to consider this data: A≤45, B≤70, A+B≤115
the number of breakfast packages x
the number of afternoon packages y

the constraint has to be from the number of A's and B's
(that is a key idea)
if we have x packages, with ⅓ lbs of A and ⅔ lbs of B
then we use x/3 lbs of A (and 2x/3 lbs of B)

similarly y packages means y/2 lbs of A and y/2 lbs of B

we add up the number of pounds of A used (when we have x of breakfast and y of afternoon): x/3 +y/2 which must be ≤ 45

similarly for B

Answer 35

x * ⅓ + y * ½ is unknown, because we not yet find how much breakfast blend should be.

2x/3 + y/2 is unknown because we not yet find how much afternoon blend should be.

Answer 36

***I think there is some error to our constraints, because:
1/3x + 1/2x ≤ 45 --> total Type A
***
I assume that is a typo, and you mean
x/3 + y/2 ≤ 45

Answer 37

ahhh I see it now! Thank you so much :))

Answer 38

yes, x and y are variable

but, looking at just the constraint caused by A.
if we use x lbs of breakfast, that means we used x/3 lbs of type A
and y lbs of afternoon means we used y/2 lbs of A
and together,
x/3 + y/2 ≤ 45 (max amount of A)
we plot the line
x/3 + y/3 = 45
or, if you like
3x + 2y = 270

Answer 39

Please hold on, I'm reading your explanation =), I'm trying to get the ideas, very tricky. :P

Answer 40

Mr.Phi, how did you get "3x + 2y = 270" please..

Answer 41

that should be 2x + 3y = 270!
x/3 + y/2 = 45

multiply by both sides (all terms) by 6 to "clear the denominators"

Answer 42

Oh jahhh! I get it =)

Answer 43

Okay, I'm trying to solve the rest of the problem. Thanks Mr.Phi.If i have anymore questions regarding to this problem, I will ask you. You're the best!

Answer 44

ty

Answer 45

Thanks Phi so much for this honorable medal..

Answer 46

I really want to reward you Mr.Phi. Please tell me how can i help you to pay back.

Answer 47

2x + 3y ≤270
4x +3y ≤420
\[x + y \le 115\]

I plugged this in my calculator, is it correct Mr.Phi?
@phi

Answer 48

yes, looks good. But there is no reason to label points outside the feasible region. You only need (0,90), (75,40), (105,0) and (maybe) (0,0) as the intersections points of the constraint lines.

Answer 49

Yah, you are right. Okay thanks!

Answer 50

(x, y) P = 1.50x + 2.00y
(0, 90) P = 1.50(0) + 2.00(90) = 180
(75, 40) P =1.50(75) + 2.00(40) = 192.50 -->Maximize profit!
(105, 0) P =1.50(105) + 2.00(0) = 157.5


Yay!!! I did it @phi helped me...

Answer 51

yes. It takes practice (for me, anyways) to be able to solve these problems quickly.

Answer 52

=P