A publisher prints and sells both hardcover and paperback copies of the same book. Two machines M and N are needed to manufacture these books. To produce one hardcover copy,machine M works 1/6 hours and N works 1/12 hours. For a paperback copy M and N work 1/15 and 1/16 hours respectively. Each machine maybe operated no more than 12 hours per day. If the profit is R12 on a hardcover copy and R8 on a paperback copy, how many of each type should be made per day to maximize the profit?
To maximize profit in this scenario, we need to determine the number of hardcover and paperback copies that should be produced per day. Let's assume the number of hardcover copies produced per day is denoted by H, and the number of paperback copies produced per day is denoted by P.
First, we need to establish the constraints for machine M and machine N. Since each machine can operate for a maximum of 12 hours per day, we can write the following inequalities:
Machine M: (1/6)H + (1/15)P ≤ 12
Machine N: (1/12)H + (1/16)P ≤ 12
Next, we need to consider the objective function, which is the profit. The profit for each hardcover copy is R12, and the profit for each paperback copy is R8. Therefore, the total profit can be calculated as follows:
Total Profit (R) = 12H + 8P
To solve this problem, we can use a technique called linear programming. We will convert the problem into a standard form and graph it to find the maximum profit.
Let's multiply all the equations by the least common multiple (LCM) of the denominators to eliminate fractions:
6(1/6)H + 6(1/15)P ≤ 6(12)
2H + P ≤ 72
12(1/12)H + 12(1/16)P ≤ 12(12)
H + (3/4)P ≤ 12
Now, let's graph the feasible region:
Graph the line: 2H + P = 72
Plot the points (0, 72) and (36, 0) and draw the line passing through them.
Graph the line: H + (3/4)P = 12
Plot the points (0, 12) and (12, 0) and draw the line passing through them.
The closed region formed by the intersecting lines is the feasible region for this problem.
Next, we need to calculate the coordinates of the vertices of the feasible region. We can do this by solving the equations formed by the intersections of the lines.
Solve the system of equations:
2H + P = 72
H + (3/4)P = 12
The solutions are H = 24 and P = 48.
Solve the system of equations:
H + (3/4)P = 12
P = 0
The solutions are H = 12 and P = 0.
Solve the system of equations:
2H + P = 72
P = 0
The solutions are H = 36 and P = 0.
Now, evaluate the profit function at each vertex:
Profit at (H, P) = 12H + 8P
Profit at (24, 48) = 12(24) + 8(48) = 288 + 384 = R672
Profit at (12, 0) = 12(12) + 8(0) = 144 + 0 = R144
Profit at (36, 0) = 12(36) + 8(0) = 432 + 0 = R432
Therefore, to maximize profit, the publisher should produce 24 hardcover copies and 48 paperback copies per day.