Engineering Mathematics IIB, MATHS 2202 Assignment 3 Due: 12:00 noon Tuesday 5 September (in the...

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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 paraboloid z = x 2 + y 2 and the plane 2x + 4y + z = 4. [3 marks] 2. Consider the plane curve x(t) = (2 cost + co
	
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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 0t 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...

Robert · Accepted Answer

1. Substituting 22 yxz   in the equation of plane, we have 
442 22  yxyx  
This can be re-written as 
9)44()12( 22  yyxx  
222 3)2()1(  yx  
This is the equation of circle with center at (-1,-2) and radius 3 units 
So, it’s parametric equation is  
1cos3  x  
2sin3  y  
2. )2sinsin2,2coscos2()( tttttx   
(i) Plot
(ii) On seeing the plot, it is seen to have 3 equal parts. So, length of curve can be given as =



3
2
0
223 dydx  
dtttdy
dtttdx
)2sin2cos2(
)2sin2sin2(


 
2222
2222
)2coscos82cos4cos4(
)2sinsin82sin4sin4(
dtttttdy
dtttttdx


 
So,

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...

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