A manufacturer produces two models of mountain bikes. Model A requires 5 hours of assembly time and 2 hours of painting time, and Model B requires 4 hours of assembly time and 3 hours of painting time. The maximum total weekly hours available in the assembly department and the painting department are 200 hours and 108 hours, respectively. The profits per unit are $25 for Model A and $15 for Model B. How many of each type should be produced to maximize profit?

Question

A manufacturer produces two models of mountain bikes. Model A requires 5 hours of assembly time and 2 hours of painting time, and Model B requires 4 hours of assembly time and 3 hours of painting time. The maximum total weekly hours available in the assembly department and the painting department are 200 hours and 108 hours, respectively. The profits per unit are $25 for Model A and $15 for Model B. How many of each type should be produced to maximize profit?

bikes of Model A

bikes of Model B

What is the maximum profit? $

Answer 1

I get 40 of A, 0 of B

for profit = 1000

maximize
25 x + 15 y
constraints
5x+4y<=300
2x+3y<=108
x>=0
y>=0
x+y>=1 (phony to fill fields)

Answer 2

http://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&id=1e692c6f72587b2cbd3e7be018fd8960&title=Linear%20Programming%20Calculator&theme=blue

try that while I look at your problem

Answer 3

Appericate it !

Answer 4

A farmer is going to divide her 40 acre farm between two crops. Seed for crop A costs $10 per acre. Seed for crop B costs $20 per acre. The farmer can spend at most $500 on seed. If crop B brings in a profit of $310 per acre, and crop A brings in a profit of $120 per acre, how many acres of each crop should the farmer plant to maximize her profit?

Answer 5

To solve this problem, we need to use linear programming to determine the optimal number of bikes of each model to produce in order to maximize profit.

Let's define the decision variables:
Let x be the number of bikes of Model A to produce.
Let y be the number of bikes of Model B to produce.

Now let's set up the constraints:
1. Assembly time constraint: 5x + 4y ≤ 200 (since the maximum total weekly hours available in the assembly department is 200 hours)
2. Painting time constraint: 2x + 3y ≤ 108 (since the maximum total weekly hours available in the painting department is 108 hours)
3. Non-negativity constraint: x ≥ 0, y ≥ 0 (we cannot produce negative quantities of bikes)

Next, we need to define the objective function, which is to maximize profit:
Objective function: Profit = 25x + 15y

Now, let's solve this linear programming problem.

Step 1: Graph the constraints on a coordinate plane.

For the assembly time constraint: 5x + 4y ≤ 200, rearranging it in the form of y = mx + b, we get:
y ≤ -5/4x + 50

For the painting time constraint: 2x + 3y ≤ 108, rearranging it in the form of y = mx + b, we get:
y ≤ -2/3x + 36

Step 2: Shade the feasible region, which is the area that satisfies all the constraints.

Step 3: Identify the corner points of the feasible region.

The corner points are the vertices of the feasible region, where the optimal solution lies.

Step 4: Evaluate the objective function at each corner point to determine the maximum profit.

Corner Point 1: (0, 0)
Profit = 25(0) + 15(0) = 0

Corner Point 2: (0, 36)
Profit = 25(0) + 15(36) = 540

Corner Point 3: (40, 20)
Profit = 25(40) + 15(20) = 1300

Corner Point 4: (38, 24)
Profit = 25(38) + 15(24) = 1290

Therefore, the maximum profit is $1300.

To determine the quantities to produce for each model, refer to the corner point with the maximum profit. In this case, the optimal solution is to produce 40 bikes of Model A and 20 bikes of Model B.