Inequalities and linear programming Examples

ExamplesA merchant plans to build a hotel with x single rooms and y double rooms. It is given that the total number of rooms should not be more than 180, while the number of single rooms should not be less than 40 and the number of double rooms should be at least 2 times the number of single rooms.
(a) Write down all the constraints about x and y.
(b) Represent the feasible solutions on a rectangular coordinate plane. 
Solutions(a) The constraints are \(\left\{ \begin{array}{l}x + y \le 180\\x \ge 40\\y \ge 2x\\x{\rm{ a n d }}y{\rm{ a r e n o n}}{\kern 1pt} {\rm{  n e g a t i v e i n t e g e r s}}\end{array} \right.\).
(b)[graphMissing Inequalities and linear programming 02 Q25]
The ordered pairs representing all points with integral coordinates in the shaded region represent all feasible solutions.

Examples(a) Draw the feasible region which represents the following constraints on a rectangular coordinate plane.
\(\left\{ \begin{array}{l}x + 3y  18 \le 0\\4x + 5y  40 \ge 0\\x \ge 6\\x{\rm{ a n d }}y{\rm{ a r e n o n}}{\kern 1pt} {\rm{  }}\,{\rm{n e g a t i v e i n t e g e r s}}\end{array} \right.\)
(b) Using the result of (a), find the maximum and minimum values of \(f(x,{\rm{ }}y) = 3x  y\) subject to the above constraints. 
Solutions(a)[graphMissing Inequalities and linear programming 02 Q42]
The dots in the graph represent all feasible solutions.
(b) Draw the straight line \(f(x,{\rm{ }}y) = 0\) on the graph in (a).
From the graph, \(f(x,{\rm{ }}y)\) attains its maximum / minimum values at the points (6, 4) and (18, 0).
At the point (6, 4), \(f(6,{\rm{ }}4) = 3(6)  4 = 14\).
At the point (18, 0), \(f(18,{\rm{ }}0) = 3(18)  0 = 54\).
∴ Maximum value\( = \underline{\underline {54}} \)
Minimum value\( = \underline{\underline {14}} \)

ExamplesAlex plans to spend $5 000 to buy x chairs A and y chairs B, where the number of chairs B should not be less than the number of chairs A, and it should not be more than twice the number of chairs A. Given that the selling prices of each chair A and each chair B are $150 and $50 respectively,
(a) write down all the constraints about x and y.
(b) represent the feasible solutions on a rectangular coordinate plane.
(c) If Alex wants to buy as many chairs as possible, how many chairs of each type should he buy? 
Solutions(a) The constraints are
\(\left\{ \begin{array}{l}150x + 50y \le 5{\rm{ }}000\\y \ge x\\y \le 2x\\x{\rm{ a n d }}y{\rm{ a r e n o n}}{\kern 1pt} {\rm{  n e g a t i v e i n t e g e r s}}\end{array} \right.\),
which are equivalent to
\(\left\{ \begin{array}{l}3x + y \le 100\\y \ge x\\y \le 2x\\x{\rm{ a n d }}y{\rm{ a r e n o n}}{\kern 1pt} {\rm{  n e g a t i v e i n t e g e r s}}\end{array} \right.\).
(b) [graphMissing Inequalities and linear programming 02 Q50]
The ordered pairs representing all points with integral coordinates in the shaded region represent all feasible solutions.
(c) Total number of chairs \(f(x,{\rm{ }}y) = x + y\)
Draw the straight line \(x + y = 0\) on the graph in (b).
From the graph, the total number of chairs is the maximum when \(x = 20\) and \(y = 40\).
∴ Alex should buy 20 chairs A and 40 chairs B.