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
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
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.
To solve this problem, we can use linear programming techniques.
Let's define two variables:
Let x = the number of bottles of cola to buy
Let y = the number of bottles of lemonade to buy
Based on the given information, we can set up the following constraints:
1. The total cost of the drinks should not exceed #1050:
50x + 40y <= 1050
2. The number of bottles cannot exceed 24:
x + y <= 24
3. The number of bottles should be non-negative:
x >= 0
y >= 0
Next, we need to maximize the profit.
Profit per bottle of cola = #10
Profit per bottle of lemonade = #5
The objective function can be defined as follows:
Profit = 10x + 5y
Now, let's graph the feasible region based on the above constraints:
1. Graph the line 50x + 40y = 1050:
To do this, we can find two points on this line using intercepts or by selecting arbitary x and y values and drawing the line passing through those points.
Let's choose x = 0, then y = 26.25 which is not possible, so let's choose y = 0, then x = 21.
So, two points on the line are (0, 26.25) and (21, 0).
Plotting these points and connecting them with a straight line will give us the line 50x + 40y = 1050.
2. Graph the line x + y = 24:
To do this, we can find two points on this line using intercepts or by selecting arbirtary x and y values and drawing the line passing through those points.
Let's choose x = 0, then y = 24 and vice versa. So, two points on the line are (0, 24) and (24, 0).
Plotting these points and connecting them with a straight line will give us the line x + y = 24.
3. Draw the x and y axes and label them accordingly.
Now, we need to find the region that satisfies all the given conditions.
The feasible region is the area enclosed by the two lines and bounded by the x and y axes.
To find the maximum profit, we need to evaluate the objective function (Profit = 10x + 5y) at each corner point of the feasible region.
The corner points of the feasible region are the points where the two lines intersect:
1. (0, 24)
2. (15, 9)
3. (21, 3)
4. (24, 0)
We can substitute these corner points into the objective function to find the profit for each combination of cola and lemonade bottles.
Profit at (0, 24) = 10(0) + 5(24) = #120
Profit at (15, 9) = 10(15) + 5(9) = #195
Profit at (21, 3) = 10(21) + 5(3) = #225
Profit at (24, 0) = 10(24) + 5(0) = #240
From the above calculations, it can be observed that the maximum profit is obtained at (24, 0). This means the woman should buy 24 bottles of cola and 0 bottles of lemonade in order to maximize the profit.
Therefore, the woman should buy 24 bottles of cola and 0 bottles of lemonade to make the maximum profit.