Solve the following questions using a calculator

Question 1

A water station gets a water delivery weekly. The volume of water sold in a week, X, is a

random variable with probability density function below (also represented on the graph).

f(x) = 5(1-x)4

, 0≤x≤1

The volume sold in week X is in units of 10 000 liters.

1a) Using the correct units, find:

– expected value of amount of water sold in a week

– variance of amount of water sold in a week

1b) In the nearest liter, what is the storage capacity that would ensure a 1% chance of the

water running out in a given week?

1c) As water is delivered, it is pumped through a storage tank.

Rate of change of water level in tank, h(t) (measured in meters) at time t (measured in

minutes) is given by this function

ℎ!

(�) = 5

2� + 3

Find height of the storage tank if it takes 20 minutes to fill.

Question 2: the displacement (represented by x in cm) of a mass on the end of a river is:

x(t)= 3�”# sin(�) ,�≥0 (time in seconds)

Question 2 (a) When does the mass first return to the starting position at x=0?

Question 2 (b) Write down an expression for the velocity of the mass

Question 2(c) Find the displacement of the mass when it first changes the direction

Question 2(d) The mass is said to have stopped oscillating when the amplitude of the

oscillation A(t)= 3�”#drops to 0.01cm. Find how long it takes for the river to stop oscillating.