Understanding the Problem: Linear Programming and Objective Functions

The question involves a concept from Linear Programming, specifically dealing with an objective function $Z = ax + by$ that is being maximized. We are given that the maximum value of this function occurs at two distinct points, (8, 2) and (4, 6), within the feasible region. We are also given constraints on the coefficients $a$ and $b$: $a \ge 0$, $b \ge 0$, and $ab = 25$. Our goal is to find the maximum value of the function $Z$.

Key Concept: Maximum at Multiple Points in Linear Programming

In Linear Programming, if an objective function attains its maximum (or minimum) value at two distinct corner points of the feasible region, then it attains the same maximum (or minimum) value at every point on the line segment connecting these two corner points. This occurs when the objective function line is parallel to the line segment connecting the two points.

This means the slope of the objective function line $ax + by = Z_{max}$ must be equal to the slope of the line segment connecting the points (8, 2) and (4, 6).

Calculating the Slope of the Line Segment

The slope of a line segment connecting two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points (8, 2) and (4, 6):

$(x_1, y_1) = (8, 2)$

$(x_2, y_2) = (4, 6)$

$\text{Slope of segment} = \frac{6 - 2}{4 - 8} = \frac{4}{-4} = -1$

Determining the Relationship between 'a' and 'b'

The objective function is $Z = ax + by$. To find the slope of the objective function line $ax + by = Z$, we can rearrange it into slope-intercept form ($y = mx + c$):

$by = -ax + Z$

$y = -\frac{a}{b}x + \frac{Z}{b}$

The slope of the objective function line is $-\frac{a}{b}$.

Since the objective function is maximum at all points on the segment connecting (8, 2) and (4, 6), its slope must equal the slope of this segment:

$-\frac{a}{b} = -1$

This simplifies to:

$\frac{a}{b} = 1$

$a = b$

Finding the Values of 'a' and 'b'

We are given the condition $ab = 25$. We have also found that $a = b$. Substitute $a$ for $b$ in the equation $ab = 25$:

$a \cdot a = 25$

$a^2 = 25$

Taking the square root of both sides:

$a = \pm \sqrt{25}$

$a = \pm 5$

We are given the constraint $a \ge 0$. Therefore, we must choose the positive value:

$a = 5$

Since $a = b$, we also have:

$b = 5$

These values ($a=5, b=5$) satisfy the conditions $a \ge 0$, $b \ge 0$, and $ab = 5 \times 5 = 25$.

Calculating the Maximum Value of the Objective Function

Now that we have the values of $a$ and $b$, the objective function is $Z = 5x + 5y$.

To find the maximum value, we can evaluate $Z$ at either of the given points where the maximum occurs, (8, 2) or (4, 6).

Evaluating at point (8, 2):

$Z = 5(8) + 5(2)$

$Z = 40 + 10$

$Z = 50$

Evaluating at point (4, 6):

$Z = 5(4) + 5(6)$

$Z = 20 + 30$

$Z = 50$

Both points yield the same maximum value.

The maximum value of the function is 50.

Revision Table: Key Steps to Solve Linear Programming Problem

Additional Information: Linear Programming Fundamentals

Linear Programming (LP) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Key components include:

• Objective Function: A linear function ($Z = ax + by$) that you want to maximize or minimize. The coefficients 'a' and 'b' represent the per-unit contribution of variables 'x' and 'y' to the objective.
• Decision Variables: The variables (like 'x' and 'y' here) whose values you need to determine to achieve the optimal solution.
• Constraints: A set of linear inequalities or equations that the decision variables must satisfy. These define the feasible region. In this problem, the constraints defining the feasible region are not explicitly given, but the points (8, 2) and (4, 6) are stated to be points where the maximum occurs, implying they are likely corner points of the feasible region. Also, $a \ge 0$ and $b \ge 0$ can be considered constraints on the parameters.
• Feasible Region: The set of all possible points $(x, y)$ that satisfy all the constraints. This region is always a convex polygon in 2D.
• Corner Points (Vertices): The vertices of the feasible region. The optimal solution (maximum or minimum) of a linear objective function always occurs at one of these corner points. If it occurs at two adjacent corner points, it occurs at all points on the edge connecting them.

This problem highlights the property that if optimality occurs at two vertices, the objective function's contour line at the optimal value is parallel to the edge connecting these vertices.