Problem

The feasible region, $R$, of a linear programming problem is shown in Figure 4.

The boundaries form part of the feasible region.

The regions excluded from the feasible region have been shaded.

Given that one of the constraints is

$3x + 2y \geq 48$

(a) state the remaining four constraints.

The five vertices of the feasible region are labelled A, B, C, D and E.

An objective function is of the form

$P = ax + by$

where $a$ and $b$ are integers.

This objective function has

• a minimum value of $\frac{202}{5}$ at vertex E
• a maximum value of $\frac{576}{3}$ at vertex C

(b) Determine the value of $a$ and the value of $b$, making your working clear.

A different objective function is of the form

$Q = x + ky$

where $k$ is a positive constant.

This objective function has a minimum value at vertex A and a maximum value at vertex C.

(c) Determine the range of values of $k$, making your working clear.