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. $8b + 4 = a$

Answer 1

Answer: (B)

$a = 2b$

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 $B$:

Slope of $AB$ is given by:

$\text{Slope of } AB = \frac{0 - 8}{4 - 0} = -2$

Thus, the ratio $\frac{a}{b} = 2$ implies $a = 2b$.