2 A landscape gardener is designing a garden. Part of the garden will be decking, part will be flowers and the rest will be grass. Let d be the area of decking, f be the area of flowers and g be the area of grass, all measured in \(\mathrm { m } ^ { 2 }\). The total area of the garden is \(120 \mathrm {~m} ^ { 2 }\) of which at least \(40 \mathrm {~m} ^ { 2 }\) must be grass. The area of decking must not be greater than the area of flowers. Also, the area of grass must not be more than four times the area of decking. Each square metre of grass will cost \(\pounds 5\), each square metre of decking will cost \(\pounds 10\) and each square metre of flowers will cost \(\pounds 20\). These costs include labour. The landscape gardener has been instructed to come up with the design that will cost the least.

1. Write down a constraint in d , f and g from the total area of the garden.
2. Explain why the constraint \(\mathrm { g } \leqslant 4 \mathrm {~d}\) is required.
3. Write down a constraint from the requirement that the area of decking must not be greater than the area of flowers.
4. Write down a constraint from the requirement that at least \(40 \mathrm {~m} ^ { 2 }\) of the garden must be grass and write down the minimum feasible values for each of \(d\) and \(f\).
5. Write down the objective function to be minimised.
6. Write down the resulting LP problem, using slack variables to express the constraints from parts (ii) and (iii) as equations.
   (You are not required to solve the resulting LP problem.)