Question

The corner points of the feasible region determined by $x + y \leq 8$, $2x + y \geq 8$, $x \geq 0$, $y \geq 0$ are $A(0, 8)$, $B(4, 0)$, and $C(8, 0)$. If the objective function $Z = ax + by$ has its maximum value on the line segment $AB$, then the relation between $a$ and $b$ is:

• A. $8a + 4 = b$
• B. $a = 2b$
• C. $b = 2a$
• D. $8b + 4 = a$

Hint:

When dealing with linear optimization problems, the slope of the constraint line often provides a critical relationship between the coefficients of the objective function. Make sure to relate the slope of the constraint line to the ratio of coefficients for finding optimal values.

Answer 1

The Correct Option is B

To find the relationship between $a$ and $b$ where the objective function $Z = ax + by$ achieves its maximum on the line segment $AB$, we analyze the vertices $A(0, 8)$, $B(4, 0)$, and $C(8, 0)$. On segment $AB$, the equation of the line can be determined. The slope of line $AB$ is given by:

$$\text{slope of } AB = \frac{0 - 8}{4 - 0} = -2.$$Hence, the line equation is $y = -2x + 8$. Consider the objective function $Z = ax + by$, rewrite it as $Z = x(a - 2b) + 8b$.
The maximum or minimum occurs along $AB$ if $a - 2b = 0$.

Therefore, $a = 2b$ is the condition for $Z$ to have a constant (maximum or minimum) value along line $AB$.

Answer 2

The line segment \(AB\) has the points \(A(0, 8)\) and \(B(4, 0)\). The objective function \(Z = ax + by\) will have a maximum value on \(AB\) if \(\frac{a}{b} = -\frac{\text{change in } y}{\text{change in } x}\).

Between points \(A\) and \(B\):

Step 1: Calculate the slope of the line segment \(AB\).

The formula for the slope of a line is given by:

\[ \text{Slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the points \(A\) and \(B\). For the points \(A(0, 8)\) and \(B(4, 0)\), we have: \[ \text{Slope of } AB = \frac{0 - 8}{4 - 0} = \frac{-8}{4} = -2 \]

Step 2: Relating the slope to the objective function.

The objective function \(Z = ax + by\) has a maximum value when the ratio \(\frac{a}{b}\) matches the negative of the slope of the line segment \(AB\), i.e., \[ \frac{a}{b} = -\text{Slope of } AB \] Substituting the value of the slope: \[ \frac{a}{b} = -(-2) = 2 \]

Step 3: Expressing the relationship between \(a\) and \(b\).

From the equation \(\frac{a}{b} = 2\), we can express \(a\) in terms of \(b\): \[ a = 2b \]

Conclusion: The objective function will have a maximum value on the line segment \(AB\) if \(a = 2b\).