A woman plans to buy some bottle of soft drinks.A bottle of Miranda is £ 50 and a bottle of sprite is £40 and she does not have more than £1050. The woman's create cannot hold more than 24 bottles.1.write down the inequalities which must be satisfied .2. Draw the graphs of the above conditions and label the region that satisfied them.3. The woman expects to make a profit of £10 per bottle of Miranda and £5 per bottle of sprite, find how many of each type should she buy in order to make the maximum profits?
50m + 40s <= 1050
m+s <= 24
graph the region, and evaluate 10m+5s at each vertex
1. Let's define the variables:
Let x be the number of bottles of Miranda.
Let y be the number of bottles of Sprite.
The inequalities that must be satisfied are:
50x + 40y ≤ 1050 (Total cost cannot exceed £1050)
x + y ≤ 24 (Total number of bottles cannot exceed 24)
2. Graph:
To plot the graph, let's solve for y in each inequality and then plot them on a coordinate plane.
50x + 40y ≤ 1050
40y ≤ -50x + 1050
y ≤ (-50/40)x + (1050/40)
y ≤ (-5/4)x + 26.25
x + y ≤ 24
y ≤ -x + 24
After plotting these two lines on a graph, you will find that they form a triangular region in the first quadrant. Label this region as the feasible region.
3. Maximum Profit:
To find the maximum profit, we need to consider the profit for each type of soft drink and maximize it.
Profit per bottle of Miranda = £10
Profit per bottle of Sprite = £5
Let P represent the total profit.
P = 10x + 5y
To maximize P within the feasible region, we need to find the corner points of the feasible region, calculate the value of P at each corner point, and select the one with the maximum profit.
By solving the inequalities, you will find the corner points:
(0,0), (15,9), and (24,0)
Now substitute each of these into the profit equation P = 10x + 5y and calculate the profit:
P(0,0) = 10(0) + 5(0) = £0
P(15,9) = 10(15) + 5(9) = £195
P(24,0) = 10(24) + 5(0) = £240
Therefore, to make the maximum profit, the woman should buy 24 bottles of Miranda and 0 bottles of Sprite.
1. Inequalities:
Let x be the number of bottles of Miranda and y be the number of bottles of Sprite.
The cost inequality can be written as:
50x + 40y ≤ 1050
The capacity inequality can be written as:
x + y ≤ 24
2. Graph:
To draw a graph of these inequalities, first, we need to convert them into equations.
The cost equation is:
50x + 40y = 1050
The capacity equation is:
x + y = 24
Let's plot these equations on a graph:
On the x-axis, plot the values from 0 to 24.
On the y-axis, plot the values from 0 to 24.
For the cost equation:
Let x = 0, y = (1050/40) = 26.25
Let y = 0, x = (1050/50) = 21
So the cost equation line starts from (0, 26.25) and ends at (21, 0).
For the capacity equation:
Let x = 0, y = 24
Let y = 0, x = 24
So the capacity equation line starts from (0, 24) and ends at (24, 0).
Draw both lines on the same graph. The region where both conditions are satisfied is the shaded region below or on the lines.
3. Maximizing Profits:
To maximize the profit, we need to maximize the total profit equation.
Total profit equation:
P(x, y) = 10x + 5y
Let's label the corner points of the shaded region on the graph as A, B, C, and D.
Calculate the profit at each of these points:
At point A (0, 24):
P(A) = 10(0) + 5(24) = 120
At point B (12, 12):
P(B) = 10(12) + 5(12) = 180
At point C (15, 9):
P(C) = 10(15) + 5(9) = 195
At point D (21, 3):
P(D) = 10(21) + 5(3) = 225
The maximum profit is achieved at point D (21, 3) with a profit of £225. Therefore, the woman should buy 21 bottles of Miranda and 3 bottles of Sprite to make the maximum profit.
To find the maximum profit, we need to set up the inequalities, draw the graph, and then find the optimal solution on the graph.
1. Inequalities:
Let's assume the woman plans to buy x bottles of Miranda and y bottles of Sprite.
Since she cannot spend more than £1050:
50x + 40y <= 1050
Also, her crate cannot hold more than 24 bottles:
x + y <= 24
2. Graph:
To draw the graph, we need to make each inequality an equation and solve for y:
a) 50x + 40y = 1050:
Solving for y, we get:
y = (1050 - 50x) / 40
b) x + y = 24:
Solving for y, we get:
y = 24 - x
Now, we can draw the graph using these equations as lines. The x-axis represents the number of bottles of Miranda (x), and the y-axis represents the number of bottles of Sprite (y).
On the graph, label the region that satisfies both inequalities.
3. Maximum Profits:
To find the number of each type she should buy to maximize profits, we need to find the coordinates of the vertex (the highest point) within the labeled region on the graph.
At this point, the woman will make the maximum profit. To do this, we need to set up a profit equation:
Profit = (profit per bottle of Miranda * number of bottles of Miranda)
+ (profit per bottle of Sprite * number of bottles of Sprite)
Given that the profit per bottle of Miranda is £10 and the profit per bottle of Sprite is £5, the profit equation becomes:
Profit = (10x) + (5y)
Now, substitute the coordinates of the vertex into the profit equation to find the maximum profit.