10

10. [Maximum mark: 15]

A shop sells chocolates. The weight, in kilograms, of chocolates bought by a random customer can be modelled by a continuous random variable $X$ with probability density function $f$ defined by

$f(x) = \begin{cases} \frac{6}{85}(4 + 3x - x^2), & 0.5 \leq x \leq 3 \\ 0, & \text{otherwise} \end{cases}$

(a) Find the mode of $X$. [2]

(b) Find $P(1 \leq X \leq 2)$. [2]

(c) Find the median of $X$. [3]

The shop sells chocolates to customers at $25 per kilogram. However, if the weight of chocolate bought by a customer is at least 0.75 kilograms, the shop sells chocolate at a discounted rate of$24 per kilogram.

(d) Find the probability that a randomly selected customer spends at most $48. [3]

(e) Find the expected amount spent per customer. Give your answer correct to the nearest cent. [5]