A school decided to buy at least 32 bags of maize and beans. The number of bag of maize were to be more than 20 and the number of bags of beans were to be at least six . A bag of maize cost shs 2500 and a bag of beans costs shs 3500. The school had shs 100000 to purchase the maize and beans.
a)Write down all the inequalities that satisfy the above information.
b)plot the inequalities in a above on the graph and shade the unwanted region
c)Show the coordinate that will minimise cost and hence determine the minimum cost of buying maize and beans.
a) The inequalities that satisfy the above information are:
- Number of bags of maize > 20
- Number of bags of beans >= 6
- Total cost of maize and beans <= 100000
b) Plotting the inequalities on a graph:
Let x represent the number of bags of maize, and y represent the number of bags of beans.
The inequality for number of bags of maize: x > 20
This is a vertical line passing through x = 20 on the x-axis, shading all the points to the right of this line.
The inequality for number of bags of beans: y >= 6
This is a horizontal line passing through y = 6 on the y-axis, shading all the points above this line.
The inequality for total cost: 2500x + 3500y <= 100000
Rearranging this inequality to slope-intercept form: y <= (-2500/3500)x + 28.57
This is a line with slope (-2500/3500) and y-intercept 28.57, shading all the points below this line.
Shading the unwanted region:
Shade the region to the right of x = 20, above y = 6, and below the line y <= (-2500/3500)x + 28.57.
c) To minimize cost, we want to find the point in the shaded region where the line y = (-2500/3500)x + 28.57 intersects the boundaries x = 20 and y = 6.
Substituting x = 20 into the equation y = (-2500/3500)x + 28.57:
y = (-2500/3500)(20) + 28.57
y = -14.29 + 28.57
y = 14.28
So the coordinates that minimize cost are (20, 14.28).
To determine the minimum cost, we substitute these coordinates into the total cost equation:
Cost = 2500(20) + 3500(14.28)
Cost = 50000 + 49980
Cost = 99980
Therefore, the minimum cost of buying maize and beans is Shs 99980.