5 Mark wants to decorate the walls of his study. The total wall area is \(24 \mathrm {~m} ^ { 2 }\). Mark can cover the walls using any combination of three materials: panelling, paint and pinboard. He wants at least \(2 \mathrm {~m} ^ { 2 }\) of pinboard and at least \(10 \mathrm {~m} ^ { 2 }\) of panelling.
Panelling costs \(\pounds 8\) per \(\mathrm { m } ^ { 2 }\) and it will take Mark 15 minutes to put up \(1 \mathrm {~m} ^ { 2 }\) of panelling. Paint costs \(\pounds 4\) per \(\mathrm { m } ^ { 2 }\) and it will take Mark 30 minutes to paint \(1 \mathrm {~m} ^ { 2 }\). Pinboard costs \(\pounds 10\) per \(\mathrm { m } ^ { 2 }\) and it will take Mark 20 minutes to put up \(1 \mathrm {~m} ^ { 2 }\) of pinboard. He has all the equipment that he will need for the decorating jobs.
Mark is able to spend up to \(\pounds 150\) on the materials for the decorating. He wants to know what area should be covered with each material to enable him to complete the whole job in the shortest time possible.
Mark models the problem as an LP with five constraints. His constraints are:
$$\begin{aligned}
& x + y + z = 24
& 4 x + 2 y + 5 z \leqslant 75
& x \geqslant 10
& y \geqslant 0
& z \geqslant 2
\end{aligned}$$
- Identify the meaning of each of the variables \(x , y\) and \(z\).
- Show how the constraint \(4 x + 2 y + 5 z \leqslant 75\) was formed.
- Write down an objective function, to be minimised.
Mark rewrites the first constraint as \(z = 24 - x - y\) and uses this to eliminate \(z\) from the problem.
- Rewrite and simplify the objective and the remaining four constraints as functions of \(x\) and \(y\) only.
- Represent your constraints from part (iv) graphically and identify the feasible region. Your graph should show \(x\) and \(y\) values from 9 to 15 only.