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Maths Students!!

I know there a lot of you here, so don't try and hide. I'm trying to answer some questions...

Find the work done by the force field F(x,y) = x sin j + yj on particle that moves along the parabola y = x² from (-1,1) to (2,4).

A 170lb man carries a 55 lb can of paint up a helical staircase that encircles a silo with a radius of 20ft. If the silo is 100 ft high and the man makes exactly three complete revolutions, how much work is done by the man against gravity in climbing to the top?

Find the area enclosed by the ellipse x²/a² + y²/b² = 1

Find the flux of the vector field F(x,y,z) = z i + y i + x k across the unit sphere x² + y² + z² = 1.

The temperature at a point in a ball with conductivity K is inversely proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere S of a radius a with center at the center of the ball.

Find the orthogonal trajectories of the family of curves for x² - y² = k and y = kx³

Solve the initial value problem for 1 + y cos xy + (x cos xy)y´ = 0, y(2) = 0

Solve the equation y´´ - 6y´ + 20y = 0

Show that there is no vector field G such that
curl G = 2xi + 3yzj – xz²k

A 20.0 lb piece of hot copper is dropped into 30.0 lb of water at 60 degree F (Fahrenheit). If the final temperature of the mixture is 85 degree F, what was the initial temperature of the copper?

For what values of x does the graph of f(x) = 2x³ – 4x² – 9x + 100 have a horizontal tangent?

A lighthouse is on a small island 8km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1.2km from P?

Find all functions g such that
g´(x) = 8 sin x – 6x^5 + 6x^¾

A wire takes the shape of the semicircle x² + y² = 1, y ≥ 0, and is thicker near its base than near the top. Find the center of mass of the wire if the linear density at any point is proportional to its distance from the line y = 1.

Now, obviously I don't want you to answer any or all of the questions for me (although that would be nice)... but because I don't recall doing any of this for A Level Maths (except maybe the derivatives and functions) can you tell me which areas of Maths I need to research into? Just short headings for each questions would be nice.

Reply 1

Pencil

Sorry Icarus, that looks like a load of jibberish to me. But I can point out that center is in actual fact spelt centre. :smile:

Reply 2

Icarus

Pencil

Sorry Icarus, that looks like a load of jibberish to me. But I can point out that center is in actual fact spelt centre. :smile:

Okay. Yeah, it's a sort of games programming course, obviously done by Americans.

Reply 3

Unregistered

1) Mechanics. Work done by a force. ( have to consider work done against gravity as well)
2) height times weight times g ( thats what i think)
3)elliptic functions ( that integral is quite easy to find)
4)Vector calculus
5)differential calculus
6)same
7)same
8)same
9)differential equations
10)thats easy
11)that is easy as well
12)that is easy. just integrate and put a constant aat the end
13)have to work that out( take time)

good luck

bloodhound

Reply 4

Icarus

Unregistered

1) Mechanics. Work done by a force. ( have to consider work done against gravity as well)
3)elliptic functions ( that integral is quite easy to find)
4)Vector calculus

good luck

bloodhound

1) Oh I see now. GUess I read force field as one word. BUt we haven't done anything on moving along a parabola, only a st. line. Oh well, better look it up.

3)Er...never even heard of it.
4) We did simple vectors, but nothing along the lines of spheres and the likes. Time for some heavy brainwork again!

Reply 5

Unregistered

here is a link.

http://mathworld.wolfram.com/

its a very good reference site. you probably find everything u want in there
good luck

bloodhound

Cheers bloodhound.

A lot of that stuff is Further Maths.

'For what values of x does the graph of f(x) = 2x³ – 4x² – 9x + 100 have a horizontal tangent? '

That is quite simple. Horizontal tangents have a gradient of 0, so you differentiate, putting it equal to 0 and find the value of x.

Reply 8

Rich

Icarus

Find the work done by the force field F(x,y) = x sin j + yj on particle that moves along the parabola y = x² from (-1,1) to (2,4).

You need to know about line integrals of vector fields.

See http://farside.ph.utexas.edu/~rfitzp/teaching/em1/lectures/node13.html and http://farside.ph.utexas.edu/~rfitzp/teaching/em1/lectures/node14.html for good explanations of line integrals of vector fields.

Icarus

A 170lb man carries a 55 lb can of paint up a helical staircase that encircles a silo with a radius of 20ft. If the silo is 100 ft high and the man makes exactly three complete revolutions, how much work is done by the man against gravity in climbing to the top?

This is a simple mass*gravity_strength*vertical_height question, the helix stuff is irrelevant, as gravity operates vertically.

Icarus

Find the area enclosed by the ellipse x²/a² + y²/b² = 1

Write the ellipse in parametric form (x = a*cos(t), y = b*sin(t)) and then integrate. You should find yourself integrating 4*a*b*sin^2(t) between 0 and pi/2 with respect to t. You should end up with the area as a*b*pi.

Icarus

Find the flux of the vector field F(x,y,z) = z i + y i + x k across the unit sphere x² + y² + z² = 1.

You need to know about surface integrals of vector fields.

See http://oregonstate.edu/dept/math/CalculusQuestStudyGuides/vcalc/flux/flux.html, http://farside.ph.utexas.edu/~rfitzp/teaching/em1/lectures/node15.html and http://farside.ph.utexas.edu/~rfitzp/teaching/em1/lectures/node16.html for good explanations of flux in relation to surface integrals of vector fields.

Icarus

The temperature at a point in a ball with conductivity K is inversely proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere S of a radius a with center at the center of the ball.

I think here you want surface integrals again (see the last question).

Icarus

Find the orthogonal trajectories of the family of curves for x² - y² = k and y = kx³

Given a family of curves, an orthogonal trajectory is a curve which intersects each member of the family at right angles (perpendicularly).

Recall for a curve to be perpendicular to another curve, their gradients must be the negative reciprocal of each other.

So for the curve family, x^2 - y^2 = k, let's differentiate:

2x - 2yy' = 0
y' = x/y

So for the orthogonal trajectory (negative reciprocals of gradients):

y' = -y/x
1/y dy = -1/x dx
ln|y| = - ln|x| + C
|y| = e^(- ln|x| + C) = A/(|x|)

So |xy| = A, where A > 0
thus xy = B, for any B, is the orthogonal trajectory family of curves.

You should be able to see how this works now.

Icarus

Solve the initial value problem for 1 + y cos xy + (x cos xy)y´ = 0, y(2) = 0

Never heard of the initial value problem, but I assume it just means to solve the above differential equation for the particular case when x = 2 and y = 0.

Icarus

Solve the equation y´´ - 6y´ + 20y = 0

A standard second-order differential equation as in P4 A-level maths.

Icarus

Show that there is no vector field G such that
curl G = 2xi + 3yzj – xz²k

You need to read up on vector fields and things like curl and divergence. See http://farside.ph.utexas.edu/~rfitzp/teaching/em1/lectures/node4.html for a nice vector calculus tutorial.

Icarus

A 20.0 lb piece of hot copper is dropped into 30.0 lb of water at 60 degree F (Fahrenheit). If the final temperature of the mixture is 85 degree F, what was the initial temperature of the copper?

Find a physics text book somewhere!

Icarus

For what values of x does the graph of f(x) = 2x³ – 4x² – 9x + 100 have a horizontal tangent?

Look for when dy/dx = 0.

Icarus

A lighthouse is on a small island 8km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1.2km from P?

Draw a little diagram and use v = r*omega circular motion equation along with some basic trigonometry...

Icarus

Find all functions g such that
g´(x) = 8 sin x – 6x^5 + 6x^¾

Basic, A-level integration.

Icarus

A wire takes the shape of the semicircle x² + y² = 1, y ≥ 0, and is thicker near its base than near the top. Find the center of mass of the wire if the linear density at any point is proportional to its distance from the line y = 1.

I believe this sort of stuff is covered in M3 centre of mass (maybe without varying density, but that shouldn't be too hard to account for), so try and get hold of an M3 textbook...

Icarus

Now, obviously I don't want you to answer any or all of the questions for me (although that would be nice)... but because I don't recall doing any of this for A Level Maths (except maybe the derivatives and functions) can you tell me which areas of Maths I need to research into? Just short headings for each questions would be nice.

I hope this has helped you some. Who asked you these questions? They seem a bit difficult for someone with no experience in the areas they involve, as a few of them are way beyond A-level maths and further maths.

Well, good luck!

Regards,

Reply 9

Unregistered

Disclaimer:i am a high school student in India(16 years)
Please correct any math or grammer mistake i might make.

Find the area enclosed by the ellipse x²/a² + y²/b² = 1

Learn multivariable calculas(now on S(0-b(1-x^2)^.5==>integral zero to b root one plus x squared .
x²/a² + y²/b² = 1
is an ellipse with centre at origin .so if we integrate total area cancels off & we get 0.So integrate in any 1 Quadrant.

area=S(0-a^2)s(0-b(1-x^2)^.5 )dxdy
dxdy is infinetisemaly samll area.
you get S(0 -a^2)(a^2-x^2)^.5.

substitute x=asin(y)
solving

Area =pi*a*b.

2)
A 20.0 lb piece of hot copper is dropped into 30.0 lb of water at 60 degree F (Fahrenheit). If the final temperature of the mixture is 85 degree F, what was the initial temperature of the copper?

heat gained by H2O is heat lost by copper.

Q(heat)=mass*specific heat capacity*temp difference

find out s of H2O & copper and calculate.

By the way Why do you need all these for?
All the questions are of different levels some simple some tough