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Chetney's RTA Woodcarving Co. manufactures two types of wooden toys. Wooden xyl...

Oct 24, 2024

Solution by Steps

step 1

Let

x

be the number of xylophones and

y

be the number of yo-yos produced. The objective function to maximize profit

P

is given by:

P = 20x + 17y - 2(x + y)

which simplifies to

P = 18x + 15y

step 2

The constraints based on labor hours are: 1. Carpentry:

2x + y \leq 40

2. Finishing:

2x + 2y \leq 50

3. Minimum xylophones:

x \geq 5

4. Non-negativity:

x \geq 0, y \geq 0

step 3

To find the feasible region, we graph the constraints on a coordinate plane. The intersection points of the constraints will help identify the feasible region

step 4

The extreme points of the feasible region can be found by solving the equations of the constraints. The points are: 1.

(5, 0)

(5, 15)

(20, 0)

(10, 10)

step 5

Evaluating the objective function

P

at each extreme point: 1. At

(5, 0)

P = 18(5) + 15(0) = 90

2. At

(5, 15)

P = 18(5) + 15(15) = 315

3. At

(20, 0)

P = 18(20) + 15(0) = 360

4. At

(10, 10)

P = 18(10) + 15(10) = 330

step 6

The optimal solution occurs at

(20, 0)

with a maximum profit of

P = 360

Answer

Chetney should manufacture 20 xylophones and 0 yo-yos for a maximum profit of RM 360.

Key Concept

Linear programming is a method to achieve the best outcome in a mathematical model with linear relationships.

Explanation

The solution involves maximizing profit while adhering to constraints on labor hours and minimum production requirements. The optimal production plan is determined by evaluating the objective function at the extreme points of the feasible region.