The daily orders for a particular product at a factory, in hundreds of kilograms, are represented by a random variable X having density function f (x) =

{

a(x − 1)2, 0 ≤ x ≤ 6, 0, elsewhere.

(i) Obtain the value of a.

(ii) Find the proportion of days in which the total amount of the product ordered is

(a) at least 450 but less than 550 kg;

(b) at most 500 kg.

(iii) Find the expected amount of the product ordered daily.