Question

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?

Answer 1

The constraints for the company are based on assembly and mechanical tuning hours: 5x + 10y ≤ 152 (assembly), 4x + 2y ≤ 23 (tuning), and x, y ≥ 0. These ensure the company stays within labor limits for mountain and racing bikes production.

Step-by-step explanation:

The constraints for the bicycle company problem are derived from the maximum available assembly and mechanical tuning hours per week, and each bicycle's time requirement for production.

1. Assembly time constraint for mountain bikes and racing bikes: 5x + 10y ≤ 152.
2. Mechanical tuning time constraint for mountain bikes and racing bikes: 4x + 2y ≤ 23.
3. Non-negativity constraints, which dictate that the company cannot produce a negative number of bicycles: x ≥ 0 and y ≥ 0.

These constraints reflect the maximum assembly and tuning labor hours, with 'x' being mountain bikes and 'y' being racing bikes.

Answer 2

The answer is below

Step-by-step explanation:

Let x represent the number of mountain bikes they produce, and let y represent the number of racing bikes they produce.

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)