6 A manufacturing company holds stocks of two liquid chemicals. The company needs to update its stock levels. The company has 2000 litres of chemical A and 4000 litres of chemical B currently in stock. Its storage facility allows for no more than a combined total of 12000 litres of the two chemicals. Chemical A is valued at \(\pounds 5\) per litre and chemical B is valued at \(\pounds 6\) per litre. The company intends to hold stocks of these two chemicals with a total value of at least \(\pounds 61000\). Let \(a\) be the increase in the stock level of A, in thousands of litres ( \(a\) can be negative).
Let \(b\) be the increase in the stock level of B , in thousands of litres ( \(b\) can be negative).

1. Explain why \(a \geqslant - 2\), and produce a similar inequality for \(b\).
2. Explain why the value constraint can be written as \(5 a + 6 b \geqslant 27\), and produce, in similar form, the storage constraint.
3. Illustrate all four inequalities graphically.
4. Find the policy which will give a stock value of exactly \(\pounds 61000\), and will use all 12000 litres of available storage space.
5. Interpret your solution in terms of stock levels, and verify that the new stock levels do satisfy both the value constraint and the storage constraint.