Question

PLEASE TYPE OUT THE ANSWERS USING TEXT ONLY SO i CAN BETTER UNDERSTAND IT, NO PICTURES PLEASE THANKS!

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 1

The objective function is maximixe Z=1.5 * x + 2 * y

x represents the morning blend.
y represents the afternoon blend.

the constraints are a maximum of 45 pounds of grade A tea and a maximum of 70 pounds of grade B tea.

the morning blend uses 1/3 of a pound of grade A and 2/3 of a pound of grade B tea.

the afternoon blend uses 1/2 of a pound of grade A and 1/2 of a pound of grade B tea.

grade A tea is therefore used at a rate of 1/3 of a pound for each pound of morning blend and 1/2 of a pound for each pound of afternoon blend.

the constraint equation for grade A tea is therefore:

1/3 * x + 1/2 * y <= 45

grade B tea is therefore used at a rate of 2/3 of a pound for each pound of morning blend and 1/2 of a pound for each pound of afternoon blend.

the constraint equation for grade B tea is therefore:

2/3 * x + 1/2 * y <= 70

now the problem looks like this

your objective function is 1.5 * x + 2.0 * y.
this represents the profit that you want to maximize.

your constraint functions are:

1/3 * x + 1/2 * y <= 45
2/3 * x + 1/2 * y <= 70

now remove inequality sign now our constraints look like

1/3 * x + 1/2 * y = 45
2/3 * x + 1/2 * y = 70

now put x=0 in both euations we get two points A=(0,70) and B=(0,140)

now put y=0 in both the equations we get C=(135,0) and D= (105,0)

now draw the graph in graph sheet

join AC and BD

you can also see that the objective function is 1.5 * x + 2 * y and the objective is to maximize the profit.

there are additional constraints that x and y have to be >= 0 because they can't be negative.

NOW THE GRAPH LOOKS LIKE THIS

your feasible region is the area on the graph that is shaded.

from the graph, it is clear that to maximize the profit she needs to blend

75 pounds of the breakfast blend and

40 pounds of the afternoon blend

maximum profit = (1.50*75)+(2*40)

=$192.50