The math help thread

just wondering what's right circular cylinder?

edit: nvm just a fancy name for cylinder >.>
working on it now

As opposed to an oblique cylinder:

or a cylinder with some other non-circular base:

The answer in the back of the book doesn't have any thetas whatsoever. And we haven't done differentiation of trigonometric functions yet. This is the fail Calculus course that people take if they couldn't take AP Calculus yet I can't solve it.

The answer says (4πr^3)/(3sqrt3) :\

Oh my, there's a figure that goes along with it.

It shows a Sphere with an inscribed Cylinder; two radii endpoints for the top and bottom bases are: (x, sqrt(r^2-x^2)) and (x, -sqrt(r^2-x^2)).

The answer in the back of the book doesn't have any thetas whatsoever. And we haven't done differentiation of trigonometric functions yet. This is the fail Calculus course that people take if they couldn't take AP Calculus yet I can't solve it.

The answer says (4πr^3)/(3sqrt3) :\

Oh my, there's a figure that goes along with it.

It shows a Sphere with an inscribed Cylinder; two radii endpoints for the top and bottom bases are: (x, sqrt(r^2-x^2)) and (x, -sqrt(r^2-x^2)).

that's the answer your going to get after you differentiate the equation and solve for theta

another way to do it 2r^2+h^2=R
and so h= sqrt(2r^2-R)
so plug that in V=PI*r^2*sqrt(2r^2-R)
differentiate and then solve for r

Now this one is considerably trickier.

To determine the new dimensions of the circumscribed rectangle based on the original rectangle's dimensions L and W, and the variable θ, we'll apply geometry and trigonometry.

Code:

sin(θ) = o/h
cos(θ) = a/h
For both cases, h will either be L or W.

X = sin(θ)*W + cos(θ)*L; new rectangle's length
Y = sin(θ)*L + cos(θ)*W; new rectangle's width

V = X*Y
  = sin(θ)^2*L*W + cos(θ)^2*L*W + sin(θ)*cos(θ)*(W^2 + L^2)
  = L*W + sin(θ)*cos(θ)*(W^2 + L^2)

We'll do the same trick as with the cone problem: find the derivative of the volume function based off of θ.

Code:

V' = (-sin(θ)^2 + cos(θ)^2)*(W^2 + L^2)

If θ = Π/4 = 45°, V' = 0.

[color=Red]V = L*W + 1/2(W^2 + L^2)[/color]

where did you get the LW?
This is how I did it.

Matt, have you tried expanding your answer instead of keeping it as factored form? They should be the same answer.

srry for such a late response xD

So, this is an annoying stat problem we were given. It's so frustrating. e.e

Sixteen passengers on a liner discover that they are an exceptionally representative body. Four are Englishmen, four are Scots, four are Irish and four are Welsh. There are also four of each of four different ages, 35,45,55,65 and no two of the same age are of the same nationality. By profession, four are lawyers, four are soldiers, four are clergymen and four doctors, and no two of the same profession are of the same age or of the same nationality.

It appears, also, that four are bachelors, four are married, four are widowed, and four are divorced and that no two of the same marital status are of the same profession, or of the same age, or of the same nationality. Finally, four are conservatives, four are liberals, four are socialists, and four are fascists, and no two of the same political sympathies are of the same marital status, or of the same profession, or of the same age or of the same nationality.

Three of the fascists are known to be a divorced English lawyer of 65, a married Scots soldier of 55 and a widowed Irish doctor of 45. It is then easy to specify the remaining fascist.

It is further given that the Irish socialist if 35, the conservative of 45 is a Scotsman and the Englishman of 55 is a clergyman. What do you know of the Welsh lawyer?

e.e

I have a mass of M. I pick up a rock of mass m and throw it into the air to land on a frozen lake. How high do I have to throw it for it to simulate me standing on the ice? Is any more information needed?
If I worded that poorly: I want to throw rocks onto a frozen lake to test the ice strength. How can I be sure it will hold me?

So, this is an annoying stat problem we were given. It's so frustrating. e.e

Sixteen passengers on a liner discover that they are an exceptionally representative body. Four are Englishmen, four are Scots, four are Irish and four are Welsh. There are also four of each of four different ages, 35,45,55,65 and no two of the same age are of the same nationality. By profession, four are lawyers, four are soldiers, four are clergymen and four doctors, and no two of the same profession are of the same age or of the same nationality.

It appears, also, that four are bachelors, four are married, four are widowed, and four are divorced and that no two of the same marital status are of the same profession, or of the same age, or of the same nationality. Finally, four are conservatives, four are liberals, four are socialists, and four are fascists, and no two of the same political sympathies are of the same marital status, or of the same profession, or of the same age or of the same nationality.

Three of the fascists are known to be a divorced English lawyer of 65, a married Scots soldier of 55 and a widowed Irish doctor of 45. It is then easy to specify the remaining fascist.

It is further given that the Irish socialist if 35, the conservative of 45 is a Scotsman and the Englishman of 55 is a clergyman. What do you know of the Welsh lawyer?

e.e

My mind sort of died a little when I read this.

HUZZAH! I GOT IT! It only took about three hours. D:

If anyone is doing it and would like to know the answer out of pure frustration, PM me.

Ugh, I'm having a really dumb moment. Got all the way through a semi long question, and now i'm having a moron moment on what should be the easiest part of the question.

How does one integrate -1/(1-x+x^2) with respect to x? I have the anwser, but can't for the life of me work out which method was used for it.
(The anwser should be

)

Matt, have you tried expanding your answer instead of keeping it as factored form? They should be the same answer.

srry for such a late response xD

Yeah, I got it, but I forgot to edit my response here.

Ugh, I'm having a really dumb moment. Got all the way through a semi long question, and now i'm having a moron moment on what should be the easiest part of the question.

How does one integrate -1/(1-x+x^2) with respect to x? I have the anwser, but can't for the life of me work out which method was used for it.
(The anwser should be

)

Multiply by 4/4:
-4 / (4-4x+4x^2) dx

Complete the square:
-4 / ((2x-1)^2+3) dx

Substitute u = 2x-1, du = 2dx:
-2 / (u^2+3) du

Inverse tan integral:
-2 arctan(u/sqrt(3)) / sqrt(3)

De-substitute u = 2x-1:
-2 arctan((2x-1)/sqrt(3)) / sqrt(3)

That should be your answer.

If you knew how many people were online in a Maple Server at any given time of the day, the distribution (let's assume it's normal) of time spent online during the day by people, and the distribution of time-of-logging in, (or approximates), how could you predict the number of unique users online over the course of a day (or smaller time frame)?

Since F=ma is pretty much the extent of my actual physics knowledge, I have a question for you all.



Air is pumped into a cylinder to push a piston, which hits a free-rotating board suspended from above, which then hits a ball. How would I figure out how far the ball can go, given the air pressure, diameter of the cylinder, weight of the ball, etc?

Pressure is Force/Area, if we assume perfectly inelastic collision between the pistol, board, and ball, then it'd just be some kind of force -> momentum conversion, then conservation of momentum. (Just divide the momentum of the board by the mass of the ball(?)). That would give you the velocity of the ball. Friction + rolling stuff -> final formula to use.
I really should pay more attention in class....

All righty. So, I really don't understand this rotation stuff that much, so it'd be much appreciated if someone were to guide me through this.

A uniform sphere of radius R and mass M rotates freely about a horizontal axis that is tangent to an equatorial plane of the sphere, as shown below. What is the moment of inertia of the sphere about this axis?



I'm assuming a 'uniform sphere' means it's solid? And also, would this qualify for the institution of the Parallel Axis Theorem? I'm not even sure.

If it is, I'd say that:

I = 2/5 M*R^2 + M*R^2 = 7/5 M*R^2 ?

Two more.

Use the Parallel-Axis Theorem to calculate the rotational inertia at the position shown in the figure below... m=20kg


From what I learned, you need to do something about making it a thin rod rotating at one end, but I'm stuck after that. Is it just the moment of inertia of the rod rotating from one end or is there something else to it?

Annd..

Using the figure below, derive an expression for the velocity of center of mass of the solid cylinder (disk) as it rolls down the incline. Also show the derivation for the linear acceleration of the center of mass.



Yeah, lost.

Ooh, ooh. I think I got the acceleration derivation for the third problem. :]

F[SIZE="1"]N[/SIZE] - f[SIZE="1"]k[/SIZE] = ma --> mgsinθ - f[SIZE="1"]k[/SIZE] = a --> gsinθ - f[SIZE="1"]k[/SIZE]/m = a
T = Iα = rf[SIZE="1"]k[/SIZE] ---> Iα/r = f[SIZE="1"]k[/SIZE]
rα=a --> α=a/r

Substitute for f[SIZE="1"]k[/SIZE]:

gsinθ - Iα/mr = a
gsinθ - mr^2α/2mr = a
gsinθ - mr^2a/2mr^2 = a
gsinθ - a/2 = a
gsinθ = 3/2a
a = 2/3*gsinθ

Can anyone confirm this?

I don't really care if I get credit for this by the deadline, but I'd like to know if I'm wrong or right. I think I'm right. I've done it a few different ways and none of them worked. Seemingly the most accurate answer is the one in the answer box.

Yep. :>



The integral of the interval (-1, 1) = 0, then you can try different Riemann Sums or (I think) add the area of the trapezoids? (1/x forms trapezoids when delta x = 1).

xleviathan:
anti derivative of 1/x =lnx
so should be ln9-ln1

and sry Hookarez can't help you, i dropped physics this term, and barely passed last term.

The integral of the interval (-1, 1) = 0, then you can try different Riemann Sums or (I think) add the area of the trapezoids? (1/x forms trapezoids when delta x = 1).

This is completely assumption since i haven't ever come across a question like this in my studies yet, but a rough solution you could do the integrals between -1..1 and 1..9, then add them, surely?

First bit is right, the -1..1 part is = 0

The actual integral of the remaining part is exactly
2*ln(3)
or
2.197224578
between the values of 1 and 9
the maths

You tried using the trapezium rule, well thats not worth doing, that unfortunately only evaluates the answer, which gives you an incorrect sum here.
Since you can evaluate the integrals definitively between the 2 values, int(1/x) = ln(x)
Between 1 and 9 this gives
ln(9)-ln(1) = ln(9/1) = ln(9) = ln(3^2)=2ln(3) by logarithm laws