A company can use two types of machine A and B, in a munifucturing plant,the number of operators required and the running cost per day are gives as write down the in equality constraints and profit function and graph the inequalities from the graph determine of machines A and B which should be used maximize profit
Let x be the number of machine A and y be the number of machine B.
The profit function can be represented as:
P(x, y) = Ax + By - Cx - Dy
where A and B are the profits made by one machine A and one machine B, respectively, in a day, and C and D are the costs for operating one machine A and one machine B, respectively.
To maximize the profit, we should minimize the costs.
Let's assume there are several constraints:
1. The number of operators required for machine A is a and for machine B is b. There are N operators available in total. Therefore, this constraint can be written as:
ax + by <= N
2. The availability of machines: There are M machines A and L machines B available. Hence, we have the following constraints:
x <= M
y <= L
3. The non-negativity constraint: Both x and y should be non-negative as a company cannot use negative numbers of machines.
x >= 0
y >= 0
So the inequality constraints are:
ax + by <= N
x <= M
y <= L
x >= 0
y >= 0
Now, to graph the inequalities and determine the optimal number of machines A and B which maximize the profit, follow these steps:
1. Draw the coordinate system with x-axis representing machine A and y-axis representing machine B.
2. Plot each of the inequality constraint lines as follows:
- ax + by = N
- x = M
- y = L
- x = 0
- y = 0
3. Shade the feasible region, which is the region satisfying all the inequality constraints.
4. Calculate the profit function (P) for each vertex (corners of the feasible region).
5. The optimal number of machines A and B is the vertex with the highest profit.