A woman plans to buy some bottles of soft drinks.a bottle of cola is #50 and a bottle of lemonade is #40 and she doesn't have more than 1050.the woman create cannot hold more than 24 bottles

Question

Draw the graph of the above condition and label the region that satisfied them
The woman expect to make a profit of #10 per bottle of cola and #5 per bottle of lemonade.find how many of each type should she buy in order to make the maximum profit

Answer 1

To begin, let's define the variables:
Let x represent the number of bottles of cola
Let y represent the number of bottles of lemonade

We are given the following conditions:
The cost of a bottle of cola is #50
The cost of a bottle of lemonade is #40
The woman doesn't have more than #1050
The woman cannot hold more than 24 bottles

Based on these conditions, we can create the following inequalities:
50x + 40y ≤ 1050 (total cost cannot exceed #1050)
x + y ≤ 24 (total number of bottles cannot exceed 24)

Let's plot these inequalities on the graph:

The first inequality, 50x + 40y ≤ 1050, can be rearranged to:
y ≤ (1050 - 50x)/40

The second inequality, x + y ≤ 24, can be rearranged to:
y ≤ 24 - x

To plot these inequalities, we can generate the following table:

x | y1 = (1050 - 50x)/40 | y2 = 24 - x
-------------------------------------
0 | 26.25 | 24
5 | 23.50 | 19
10 | 20.75 | 14
15 | 18 | 9
20 | 15.25 | 4
24 | 13.75 | 0

Plotting these points and drawing the lines, we obtain the graph as shown below:

|
26 -| y1
|
24 -| /
| /
22 -| /
| /
20 -| /
|/
18 -
|
16 -|\
| \
14 - \
| \
12 - \
| \
10 - \ y2
| \
8 - \
|______________________\
0 5 10 15 20 24

The shaded region represents the feasible region that satisfies the given conditions: the total cost is less than or equal to #1050 and the total number of bottles is less than or equal to 24.

To find the number of bottles that will maximize the profit, we need to evaluate the profit function.

The profit per bottle of cola is #10 and the profit per bottle of lemonade is #5. Therefore, the profit function f(x, y) can be defined as:

f(x, y) = 10x + 5y

We need to find the maximum value of this function within the feasible region.

Using the corner point method, we evaluate the profit function at the following points:

f(0, 24) = 10(0) + 5(24) = #120
f(5, 19) = 10(5) + 5(19) = #155
f(10, 14) = 10(10) + 5(14) = #190
f(15, 9) = 10(15) + 5(9) = #225
f(20, 4) = 10(20) + 5(4) = #240
f(24, 0) = 10(24) + 5(0) = #240

From these evaluations, we can see that the maximum profit occurs when x = 20 and y = 4. Therefore, the woman should buy 20 bottles of cola and 4 bottles of lemonade in order to make the maximum profit.