A company produces two types of bicycles; mountain bikes and racing bikes. It takes 5 hours of assembly time and 4 hours of mechanical tuning to produce a mountain bike. It takes 10 hours of assembly time and 2 hours of mechanical tuning to produce a racing bike. The company has at most 23 hours of mechanical tuning labor per week and at most 152 hours of assembly labor per week. The company's profit is $40 for each mountain bike produced and $140 for each racing bike produced. The company wants to make as much money as possible. Let x = the number of mountain bikes they produce, and let y = the number of racing bikes they produce. What are the constraints for this problem?

It takes 4 hours of mechanical tuning to produce a mountain bike and 2 hours of mechanical tuning to produce a racing bike. The company has at most 23 hours of mechanical tuning labor per week. Therefore the constraint is:

4x + 2y ≤ 23 (1)

It takes 5 hours of assembly time to produce a mountain bike and 10 hours of assembly time to produce a racing bike. The company has at most 152 hours of mechanical tuning labor per week. Therefore the constraint is:

5x + 10y ≤ 152 (2)

Also, x, y ≥ 0 (3)

The company's profit is $40 for each mountain bike produced and $140 for each racing bike produced. The profit equation is:

Profit = 40x + 140y

Plotting the constraints equation 1 and equation 2 using geogebra online graphing tool. The solution are (0,0), (5,75, 0), (0, 11.5)

At (0,0); Profit = 40(0) + 140(0) = 0

At (5.75,0); Profit = 40(5.75) + 140(0) = 230

At (0,11.5); Profit = 40(0) + 140(11.5) = 1610

The maximum profit is at (0, 11.5)