Question

Need help with one word problem!

Answer 1

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 afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?

Answer 2

@ganeshie8 can you help me with this last problem?

Answer 3

@ganeshie8 ?(:

Answer 4

this is a linear programming problem
u have some idea about it right ?

Answer 5

not really..

Answer 6

sure ? never heard of linear programming before ?
setting up a function, and maximizing it within the given linear constraints ?

Answer 7

i heard of it but never understood how to do it..

Answer 8

good :) lets do this and try to understand now

Answer 9

okay !

Answer 10

Say, she makes \(x\) packets of breakfast blend, and \(y\) packets of afternoon blend

Answer 11

since she gets $1.5 for each packets she sells on breakfast blend, and $2 on afternoon blend, Profit function wud be :-

\(\large P = 1.5x + 2y \)

Answer 12

fine so far ?

Answer 13

okay good so far..

Answer 14

next see that, we dont have abundant supply of A/B grade to make these packets.
so we need to make them carefully

Answer 15

okay

Answer 16

breakfast blend has a ratio of tea types, 1/3A : 2/3B

afternoon blend has a ratio of tea types, 1/2A : 1/2B

Also, A <= 45 and B <= 70

Answer 17

From above limitations,

\(\large x*\frac{1}{3} + y * \frac{1}{2} \le 45\)

and,

\(\large x*\frac{2}{3} + y * \frac{1}{2} \le 70\)

Answer 18

see if that makes some sense,
if it does, then we're almost done.

Answer 19

yea it does

Answer 20

first inequality above is for Grade A tea
second inequaloty above is for Grade B tea

Answer 21

sure ?

Answer 22

yeep got that

Answer 23

okay, so here is the linear programming setup :-

\(\large P = 1.5x + 2y\)

constraints :
\(\large x*\frac{1}{3} + y * \frac{1}{2} \le 45 \)

\(\large x*\frac{2}{3} + y * \frac{1}{2} \le 70\)

Answer 24

okay what we suppose to do

Answer 25

we need to graph those constraints and find the intersection points.

Answer 26

we got 3 intersection points :-
1) on y axis : x=0, y=90
2) on x axis : x=105, y=0
3) when lines cross : x=75, y = 40

Answer 27

So, one of these points WILL GIVE THE MAXIMUM value for our Profit function.

Answer 28

put each point in Profit function, and see wat values u get.

Answer 29

how do i do that

Answer 30

\(P = 1.5x + 2y \)

put first point,
1) on y axis : x=0, y=90

Answer 31

P = 180

Answer 32

thats right

test other two points also.

Answer 33

2) on x axis : x=105, y=0

Answer 34

p = 157.5

Answer 35

P = 192.5

Answer 36

profit is less than first point, so discard 2nd point.

Answer 37

So, which point has greater profit ?

Answer 38

3rd

Answer 39

Yes

3) when lines cross : x=75, y = 40

Answer 40

Making 75 breakfast blends, and 40 afternoon blends gives her MAXIMUM PROFIT.
and the maximum profit is $192.5

Answer 41

oh thats all i had to do?

Answer 42

thats all,
you saying as if its simple enough to do lol :o

Answer 43

well no it not all simple but i thought it would be more than that lol (:

Answer 44

good :) always prepare for the worse.. pessimism is always good lol

Answer 45

jk :)

Answer 46

which i always do cause i always think its going to be hard lol. Thank you for your help (:

Answer 47

np :)