Reference no: EM133058463

Question 1

A solution to the following differential equation is found using Euler's Method,

dy/dx - xy = 2

Using the initial condition of y(0) = k and two iterations the approximate solution y(1) ≈ 1.75 was found.

(a) Find the value of k.

Question 2

The following function is defined as

f(x) = √(4 - x/1+2x)

(a) State the domain and range of the function.
(b) Using a suitable method find the first three terms of the polynomial expansion for f(x).
(c) Using your expansion find an approximation to V5.

Question 3
A boat in a harbour and its cargo is continuously unload. into a warehouse between 6am and midday. The mass, M, of the cargo in the warehouse changes over time, t, (in hours) after 6am and can be modelled by the Nuation, dM/dt = √(4 - M²/5)

(a) By considering the initial conditions, find an equationf the mass of the cargo in the warehouse, t hours after 6am.

The bottom of the boat is D below the surface of the water. D .11 change as cargo is removed from the boat and can be modelled by the equation,

D = 2 - M

As the tide rises and fa. over the course of a day, the height, h, of the sea in the harbor at a, one point in time can be modeled with the equation,

h = 2.5 + cos (t/5)

(b) Let s be the vertical distance between the bottom of the boat a. the seabed. 

i. Express in terms of D and h.
ii. By using an appropriate identity on Rcos(x - α) show that acos(x) + bsin(x) can be written M the form Rcos(x - α).
iii. Hence show that s can be written M the form, where b, R and d a are constants to be found.

s = b + Rcos(t/5 - α)

iv. Hence or otherwise find the greatest distance of s and the time that it occurs between 6 am and midday.

The bottom of the boat is tethered to an anchor on the seabed by a steel cable. As s changes the cable is stretched and compressed which causes it stress. The stress on the cable is proportional to the rate of change of s.