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c. What combination of

x

and

y

will yield the optimum for this problem? Ma...

May 8, 2024

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Solution by Steps

step 1

To find the optimal combination of

x

and

y

that maximizes the function

14x + 12y

, we need to consider the given constraints

step 2

The constraints are

2x + 3y \leq 12

5x + 2y \leq 10

3x + 3y \leq 18

, and

x, y \geq 0

step 3

We can use the method of linear programming to solve this optimization problem. This involves finding the vertices of the feasible region defined by the constraints

step 4

The vertices of the feasible region can be found by solving the system of equations formed by the intersection of the constraints

step 5

After finding the vertices, we evaluate the objective function

14x + 12y

at each vertex to determine which gives the maximum value

step 6

According to the asksia-ll calculator, the maximum value of the objective function is

\frac{564}{11}

, which occurs at the point

(x, y) = \left(\frac{6}{11}, \frac{40}{11}\right)

[1] Answer

The optimal combination of

x

and

y

that maximizes the function

14x + 12y

(x, y) = \left(\frac{6}{11}, \frac{40}{11}\right)

, with a maximum value of

\frac{564}{11}

Key Concept

Linear Programming Optimization

Explanation

The key concept in solving this problem is linear programming, which is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. It involves maximizing or minimizing a linear objective function, subject to linear inequalities called constraints.