The total cost, $C, of manufacturing n batches of biscuit boxes is given by the formula C = an + b, where a and b are constants. When 200 batches of biscuit boxes are manufactured, the total cost is $55 000.
When 500 batches of biscuit boxes are manufactured, the total cost is $62 500.
(i) Find the value of a and of b.
(ii) Calculate the total cost of manufacturing 420 batches of biscuit boxes.
(iii) Find the number of batches of biscuit boxes manufactured if the total cost is $70 000.
(iv) Draw the graph of C against n. Is C directly proportional to n? Use your graph to explain your answer.