The feasible region and its corner points

The workshop of the last lesson has another limit: the varnish.
A table needs a litre of varnish, and a chair a litre too.
40 litres of varnish reach the workshop each week.
$\displaystyle x$ is the number of tables a week, and $\displaystyle y$ the number of chairs.

A production plan must now keep four constraints together:

• $\displaystyle x\ge0$, $\displaystyle y\ge0$ – the number of products is not negative;
• $\displaystyle 3x+y\le60$ – the hours of work;
• $\displaystyle x+y\le40$ – the varnish.

On the graph board each constraint shades its half-plane.

• The part shaded in all four colours together – the darkest – is the feasible region: all the plans that keep all the constraints.
• The feasible region here is a quadrilateral. Every point inside it, or on its sides, is a plan the workshop can carry out.
• A plan outside it breaks at least one constraint: $\displaystyle (5,38)$ keeps the hours constraint ($\displaystyle {15+38}={53}$), but needs 43 litres of varnish.

The vertices of the feasible region are called its corner points. On the graph board, the boundary lines: $\displaystyle {y}={60-3x}$ – the hours, and $\displaystyle {y}={40-x}$ – the varnish (the dashed one).

Each corner point is where two boundary lines meet:

• $\displaystyle {O}={(0,0)}$ – the axes: nothing is made.
• $\displaystyle {A}={(0,40)}$ – the varnish line and the $\displaystyle y$-axis: chairs only, all the varnish.
• $\displaystyle {B}={(10,30)}$ – the hours line and the varnish line: all the hours and all the varnish.
• $\displaystyle {C}={(20,0)}$ – the hours line and the $\displaystyle x$-axis: tables only, all the hours.

Not every meeting point is a corner: the hours line meets the $\displaystyle y$-axis at $\displaystyle (0,60)$, but that needs 60 litres of varnish – the point is outside the region.

At the corner $\displaystyle B$ the two boundary lines meet, so it is the solution of the system of their equations.
The example below subtracts the second equation from the first:

• The subtraction cancels $\displaystyle y$: $\displaystyle {2x}={20}$, so $\displaystyle {x}={10}$.
• We substitute in the second equation: $\displaystyle {10+y}={40}$, so $\displaystyle {y}={30}$.
• The corner: $\displaystyle (10,30)$ – 10 tables and 30 chairs.

The bakery of the last lesson bakes cakes and pies.
A cake takes the oven for 2 hours, and a pie for an hour; the oven is free for 24 hours a week.
A cake needs a kilogram of flour, and so does a pie; the bakery has 16 kilograms of flour a week.
$\displaystyle x$ is the number of cakes a week, and $\displaystyle y$ the number of pies.

Write the oven’s constraint and the flour’s constraint on the graph board.

On the graph board, the boundary lines of the two constraints: $\displaystyle {y}={24-2x}$ – the oven, and $\displaystyle {y}={16-x}$ – the flour (the dashed one).
Mark the corner points of the feasible region, apart from the origin.

A small factory sews backpacks and handbags.
A backpack takes 3 hours of sewing, and a handbag 2 hours; there are 120 hours of sewing a week.
Every bag, of either kind, needs a square metre of cloth; there are 50 square metres of cloth a week.
$\displaystyle x$ is the number of backpacks, and $\displaystyle y$ the number of handbags.
Solve the system of equations of the two boundary lines: in which plan does the factory use all the sewing hours and all the cloth?

A furniture shop has ordered at least 15 chairs a week from the workshop, so the workshop now has the constraint $\displaystyle y\ge15$ too.
On the graph board, the three boundary lines: $\displaystyle {y}={60-3x}$ – the hours, $\displaystyle {y}={40-x}$ – the varnish (the dashed one), and $\displaystyle {y}={15}$ – the order.
Mark the four corner points of the new feasible region.
The varnish line and the order line meet at the point $\displaystyle (25,15)$. Why is it not a corner point?