Please help this question is from my unit on linear programing and I'm having trouble completing it.
A company manufactures bicycles and tricycles. The total number of frames the company manufactures cannot exceed 80 per month. It takes the company 1 hour to assemble a bicycle and 2 hours to assemble a tricycle. The assembly machine is only available for 100 hours each month. If the company makes a profit of $50 on each bicycle and $70 on each tricycle, determine the number of bicycles and tricycles that will maximize profits each month. Indicate the maximum profit the company can make each month.
This is what I have so far: x=#of bicycles y=#of tricycles
x>=0 y>=0 P=50x +70y
Now I don't know what to do. Any help would be greatly appreciated. Thanks
To solve this linear programming problem, you need to set up the constraints and the objective function.
1. Constraints:
- The total number of frames (bicycles + tricycles) cannot exceed 80 per month. So, the constraint for frames is: x + y <= 80.
- The assembly machine is available for 100 hours each month. Assembling a bicycle takes 1 hour, and assembling a tricycle takes 2 hours. So, the time constraint is: x + 2y <= 100.
2. Objective function:
The objective is to maximize the profit. The profit for each bicycle is $50, and for each tricycle is $70. So, the objective function is: P = 50x + 70y.
Now, you can graph these constraints using a graphing calculator, a spreadsheet, or by hand.
Graph the constraints:
- Create an x-y coordinate plane.
- Plot the line x + y = 80. To do this, plot the points (80, 0), and (0, 80), and draw a line through them.
- Plot the line x + 2y = 100. To do this, plot the points (100, 0), and (0, 50), and draw a line through them.
- Shade the region of feasible solutions, which is the area below or on the lines.
Now, to find the maximum profit, you need to identify the corner points of the feasible region.
- Let's find the intersection point of the two lines. Solve the system of equations x + y = 80 and x + 2y = 100. Subtract the first equation from the second: (x + 2y) - (x + y) = 100 - 80. Simplifying, y = 20. Substituting this value into x + y = 80, x + 20 = 80, and x = 60. So, the intersection point is (x, y) = (60, 20).
- Examine the other corners of the feasible region to determine if they are valid solutions.
Calculate the profits at each corner point:
1. (0, 0): P = 50(0) + 70(0) = $0.
2. (0, 50): P = 50(0) + 70(50) = $3500.
3. (60, 20): P = 50(60) + 70(20) = $4000.
4. (80, 0): P = 50(80) + 70(0) = $4000.
The maximum profit occurs at the point (60, 20) with a profit of $4000 each month. This means the company should manufacture 60 bicycles and 20 tricycles to maximize profits.