Engineering Mathematics IIB, MATHS 2202 Assignment 3 Due: 12:00 noon Tuesday 5 September (in the assignment drop boxes) There are seven questions in this assignment. When presenting your solutions to...

School of Mathematical Sciences Engineering Mathematics IIB, MATHS 2202 Assignment 3 Due: 12:00 noon Tuesday 5 September (in the assignment drop boxes) There are seven questions in this assignment. When presenting your solutions to the assignment, please include some explanation in words to accompany your calculations. It is not necessary to write a lengthy description, just a few sentences to link the steps in your calculation. Messy, illegible or inadequately explained solutions may be penalised. The marks awarded for each part are indicated in brackets. 1. Find a parametric representation of the curve given by the intersection of the 2 2 paraboloid z =x +y and the plane 2x + 4y +z = 4. [3 marks] 2. Consider the plane curve x(t) = (2 cost + cos 2t; 2 sint sin 2t) for 0t 2. (a) Plot the curve. [1 marks] (b) Find the length of the curve. [4 marks] You may nd some of the following trigonometric identities useful: sin(A +B) = sinA cosB + cosA sinB; cos(A +B) = cosA cosB sinA sinB; 2 2 cos 2 = 2 cos  1 = 1 2 sin : 3. Consider a particle whose position isx(t) = 3 costi+3 sintj +4tk at timet 0. (a) What is the name of the curve that the particle is travelling along? [1 mark] (b) Find the unit tangent to the path at time t. [1 mark] (c) What is the speed of the particle? [1 mark] (d) Find the tangential and normal components of acceleration. [1 mark] Questions continue on the next page.4. The motion of a satellite of mass m is governed by the equation dv k m = r; 3 dt r where v is the velocity,t is time,k is a positive constant, r is the position of the satellite with respect to the centre of the planet and r =jrj. Show that the angular momentum h =rmv is constant once the satellite has been put into motion. [4 marks] p 2 2 5. Find a unit normal to the surface of the conez = x +y at the point (3; 4; 5). [2 marks] 6. The height of a region of land above sea level is 2 2 h(x;y) = 2000 2x 5y : At the point (2; 1): (a) What is the direction of steepest...