Problem 1
A farmer has 100 acres of land and can plant either wheat or corn. Wheat yields TZS 200,000 profit per acre and corn yields TZS 300,000 profit per acre. The farmer has 1200 hours of labor available, with wheat requiring 10 hours per acre and corn requiring 15 hours per acre. How many acres of each should be planted to maximize profit?
Solution: Let x = acres of wheat, y = acres of corn
Maximize P = 200,000x + 300,000y
Constraints:
x + y ≤ 100 (land)
10x + 15y ≤ 1200 (labor)
x ≥ 0, y ≥ 0
Optimal solution: (60, 40) with profit TZS 24,000,000
Problem 2
A company produces two products, A and B. Product A requires 2 hours of assembly and 1 hour of finishing, while product B requires 1 hour of assembly and 3 hours of finishing. The company has at most 100 assembly hours and 120 finishing hours available each day. If profit is TZS 30,000 for each A and TZS 50,000 for each B, how many of each should be produced daily to maximize profit?
Solution: Let x = units of A, y = units of B
Maximize P = 30,000x + 50,000y
Constraints:
2x + y ≤ 100 (assembly)
x + 3y ≤ 120 (finishing)
x ≥ 0, y ≥ 0
Optimal solution: (30, 30) with profit TZS 2,400,000
Problem 3
A bakery makes two types of cakes: chocolate and vanilla. Each chocolate cake requires 200g flour and 150g sugar, while each vanilla cake requires 250g flour and 100g sugar. The bakery has 10kg flour and 6kg sugar available. If profit is TZS 5,000 per chocolate cake and TZS 4,000 per vanilla cake, how many of each should be made to maximize profit?
Solution: Let x = chocolate cakes, y = vanilla cakes
Maximize P = 5,000x + 4,000y
Constraints:
200x + 250y ≤ 10,000 (flour)
150x + 100y ≤ 6,000 (sugar)
x ≥ 0, y ≥ 0
Optimal solution: (20, 24) with profit TZS 196,000
Problem 4
A furniture company makes chairs and tables. Each chair requires 5 hours of carpentry and 2 hours of finishing, while each table requires 10 hours of carpentry and 3 hours of finishing. The company has 300 carpentry hours and 120 finishing hours available per week. If profit is TZS 25,000 per chair and TZS 40,000 per table, find the optimal production mix.
Solution: Let x = chairs, y = tables
Maximize P = 25,000x + 40,000y
Constraints:
5x + 10y ≤ 300 (carpentry)
2x + 3y ≤ 120 (finishing)
x ≥ 0, y ≥ 0
Optimal solution: (24, 24) with profit TZS 1,560,000
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