4 Two products are to be made from material that is supplied in a single roll, 20 m long and 1 m wide. The two products require widths of 47 cm and 32 cm respectively. Two ways of cutting lengths of material are shown in the plans below.
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1. Given that there should be no unnecessary waste, draw one other cutting plan that might be used for a cut of length \(z\) metres.
2. Write down an expression for the total area that is wasted in terms of \(x , y\) and \(z\). All of the roll is to be cut, so \(x + y + z = 20\).
   There needs to be a total length of at least 20 metres of the material for the first product, the one requiring width 47 cm .
3. Write this as a linear constraint on the variables. There needs to be a total length of at least 24 metres of the material for the second product, the one requiring width 32 cm .
4. Write this as a linear constraint on the variables.
5. Formulate an LP in terms of \(x\) and \(y\) to minimise the area that is wasted. You will need to use the relationship \(x + y + z = 20\), together with your answers to parts (ii), (iii) and (iv).
6. Solve your LP graphically, and interpret the solution.