Problem

Figure 4 shows the constraints of a linear programming problem in $x$ and $y$, where $R$ is the feasible region.

The equations of three of the lines, and the four intersection points, $A$, $B$, $C$ and $D$, are shown.

The four vertices of $R$ are $A\left(\dfrac{27}{4}, \dfrac{27}{2}\right)$, $B(5, 5)$, $C(10, 7)$ and $D\left(\dfrac{9}{4}, \dfrac{21}{2}\right)$.

(a) Determine the inequalities that define $R$.

The objective function, $P$, is given by

$P = 2x + ky$

where $k$ is a positive constant.

Given that the minimum value of $P$ is at least half of the maximum value of $P$,

(b) determine the range of possible values of $k$.