Mystified by mathematics? Addled by addition? Perturbed by puzzles? Post a comment here with your question and I’ll do my best to answer and enlighten you. I won’t help you with your homework, however: it’s yours for a reason 
Mystified by mathematics? Addled by addition? Perturbed by puzzles? Post a comment here with your question and I’ll do my best to answer and enlighten you. I won’t help you with your homework, however: it’s yours for a reason
The Regard! The Excellent forum! Thank you!
http://danuegonax.com
Forgive that beside You was little ed!
Excellent forum, added to favorites!
http://srubibablo.com
So interesting there was that I fell asleep…
If I have a circle represented by x^2+y^2=1, it will be centered on the origin and extend 1 in all direction. If I have a circle x^2+(1/a^2)y^2=1, it will extend a on both the positive and negative parts of the y axis.
x^2+1/4y^2=1does through the points (1,2), (-1,2), (-1,-2), (1,-2). Does it have twice the area of x^2+y^2=1?
Also, please tell me the volume formula for a regular tetrahedron with side s. I stayed up until 10 one night, but can’t figure it out.
Thank you,
Xanaphia
Hi Xanaphia, and my apologies for taking so long to reply. I should have put a note on the blog about taking some time off to rest over Xmas!
Anyway, your question about circles is really one about ellipses. The ellipse x^2/a^2 + y^2/b^2 = 1 goes through the points on the axes (0,b),(0,-b),(a,0),(-a,0), which is a bit different from what you wrote. This ellipse has area \pi*a*b so in the example you gave, the answer is “yes”. There is no general closed form for the perimeter of the ellipse without resorting to elliptic integrals.
The volume of the regular tetrahedron with side s is V=(s^3)*\sqrt{2}/12.
You can work this about by starting with the formula V=(base area)*(height)/3 and going from there. Let me know if you get stuck!
Thanks. Another question continues on the theme of the tetrahedron one. Is there a way to calculate any regular polyheron’s area based on side? Is there a pattern?
Thank you again,
Xanaphia
Hi Xanaphia. A regular polyhedron is composed of regular polygons, so if you know the areas of the polygons, and how many there are, then you know the area of the polyhedron.
Sorry, mistyped that. I mean, can you get the area of a polygon from the side, so for instance the area of a nonagon with 6 cm sides.
Also, I quote V=(s^3)*\sqrt{2}/12 from your earlier post. Why is it */ ? Do you mean V=(s^3)*(1\sqrt{2}/12) or something else?
Thank you again.
Hi Xanaphia,
The answer to your first question is “yes“. (See also here.)
I apologise for my confusing typing in my previous answer. I typed \sqrt because this is how you generate a square root sign in the typesetting language LaTeX. I do not mean 1/sqrt.
Hi !
I was looking for Runge-Kutta Methods iterations and evaluations, but I haven’t found the information about it. I want to know what they are and how they affect the computational cost of the method.
Thank you very much !!!
Hi Ann, and thanks for your question.
Runge-Kutta methods are very useful numerical tools for estimating the value of a function at the next time step, given information about it at the current time step. The information takes the form of slopes at the beginning and end of the interval (from current time to next) and in the middle.
There are many R-K methods, and I highly recommend the article at Wikipedia: http://en.wikipedia.org/wiki/Runge-kutta
Thank you very much for your answer.
I have read the article, but I don’t understand the relation about evaluations and iterations yet, and what they really are. Can you specify a little bit more ?
Thank you very much again
On the website
http://mysite.du.edu/~jcalvert/math/hyperb.htm
(see under Hyperbolic Functions section) it mentions that the parameter, t, for the hyperbola in (cosh t, sinh t) is twice the swept out area. A comparison is made with the unit circle (cos t , sin t). What I fail to comprehend that in the case of the circle the parameter t is also the angle submitted, measured in radians. How can the parameter be an area and an angle simultaneously?
Hi Ann, and apologies for the slow reply. To evaluate a function means to work out its numeric value. To iterate means to repeat. R-K methods are iterative: you take your answer from step n, plug it back into the formula and get a new (better) answer at step n+1. This is the iterative process.
Hi David, and thanks for an excellent question! How can t be both an angle (which has no dimensions) and an area (which has the dimensions of length x length)? The key point is to note that general circles and hyperbolae are not being discussed, but rather specific ones. Consider the circle in the figure: it has radius 1. This radius (a length) is being squared in the formula to give the segment area, but since 1×1=1, you can’t see this length squared term in the final formula. We have, if you like, “non-dimensionalised” the circle by dividing all lengths by its radius. This is fundamental to the mathematical process, and is of special importance in applied mathematics. Indeed, the art of nondimensionalisation and scaling is incredibly powerful. There are lots of good books on this, such as by Barenblatt.
I have two charts. One chart has the likelihood that a musical pitch will occur in “good” western melodies. Each is weighted with a number. In the key of C Major the notes are weighted like this;
C=5, D=3.5, E=2, F=4, G=4.5, A=3.5, B=4.
I have another chart with the frequency of intervals that occur within “good” western melodies. They are weighted like this;
Unison=5 (the next note is the same as the previous note)
Minor Second=5.5
Major Second=15
Minor Third=2.5
Major Third=2
Perfect Fourth=3
Tritone=0
Perfect Fifth=1.5
Minor Sixth=.125
Major Sixth=.25
Minor Seventh=.25
Major Seventh=.25
Octave=1
I want to combine this data to find out the likelihood of all the possible combinations of motion. I’m looking for the answer to all of the questions below;
What is the likelihood that C will move to D? The answer would be expressed something like C has a 24 chances out of 123 to move to D.
What is the likelihood that C will move to the E above?
what is the likelihood that C will move to the E below?
What is the likelihood that Middle C will move to F above? to the F below? to G above? to G below? to A above? to A below? to B above, to B below? to the octave above to the octave below?
but we’re not finished.
What is the likelihood that D will move to the C below?
What is the likelihood that will will move to the C above?
What is the likelihood that D will stay on D?
What is the likelihood that D will move to E above? to the E below?
to F above, to the F below? to G above, to the G below? to A above, to the A below, to B above, to B below?
still not finished.
What is the likelihood that E will move to C the below? to the C above?
What is the likelihood that E will move to D above? to D below?
What is the likelihood that E will stay on E? Move to F above? F below? G above? G below, A above? A below? B above? B below? Octave above? Octave below?
Etc…
The difficulty I’m having with the question is correctly graphing the problem. I’ve tried several times and each time I notice something I missed before in order to get the right answer. Especially confusing to me is the Frequency of notes graph. I thought I could put the name of the pitches on the left hand side of the page with D above at the bottom since the Major second was the most important interval then mark up lines on the graph paper the appropriate number (15).
Then at the top of that write the word Unison since that is the second most important interval continuing on up in that manner. On the right hand side of the page I wrote C on the bottom across from the first item in the other graph. Then I counted up the appropriate number of lines. then wrote the next two common pitches E and G. continuing up like that.
Then I drew a line between C the pitch to the unison interval in the other graph looking for the point in-between that would show the likelihood that the pitch would stay on the same pitch based on what pitch your on in a melody? What I want to know is when I’m looking at the melody I’m writing and I’m in the middle of writing it and, for example, I’m on the note D, then what is the likelihood that D will move to any other note?
It has the greatest chance of going back to C since D is a less important pitch and since the motion of the Major Second is more important than the motion repeating it again. But I want to know the exact percentage of every possibility.
Tall order. I know. But it’s fun and interesting.
Hi Greg, and thanks for your long and interesting question. I’m sorry about this, but the blog is hibernating, as you may have noticed, and I don’t think I have the time to answer your question with the care it deserves.
Good luck!
A performance based company pays its best employee top pay. (M=$15.00/hr) And it’s worst employee the lowest rate. (m=$6.00/hr) The company is shooting for an average pay rate each week. (A = $8.50/hr) The number of employees can change week to week, we’ll assume 10 for this week. (E=10)
how do you figure how much to give second place and third and so on down to 10th place.
1. $15.00/hr
2. ?
3. ?
4. ?
5. ?
6. ?
7. ?
8. ?
9. ?
10. $6.00/hr
i think variance is the key but all the examples i see ask for the data set first. also i don’t know how to adjust for the $8.50/hr average.
thanks
Kyle, it depends what you mean by “average”: the mean, the mode, the median?
i’m not sure. when all hourly rates from 1st thru 10th are added up then the toal divided 10 (the number of employees). that number should equal $8.50/hr. been a while since i’ve taken any math but i always thought that was called the average?
it turns out after doing a little research that by “average” i meant “mean”.
My friend and I have been puzzled by how to calculate a working fight system for a game with given rules. Here is the problem: http://www.gamedev.net/community/forums/topic.asp?topic_id=499364
Please, if you would be so kind, leave me a post on that forum. Thank you.
I came across a description of this function years ago (I think in a differential equations textbook) and attempted to solve it on and off for a long time. This function has the characteristic that the tangent length (between the function and where the tangent line intersects the x-axis) has constant value 1. Clearly the function has value 1 and infinite slope at x=0 and the function asymptotically approaches zero as x increases to infinity.
I had an insight recently and (with the help of an online integrator function) solved for the function’s inverse:
x = -ln(y) + ln(sqrt(1-y^2)+1) – sqrt(1-y^2)
It’s an elegant looking broad spike.
Questions for you:
a) Any ideas on how to invert this to get y in terms of x?
b) Is there a common name or term for this function?
Thanks!
I use a website that helped me learn inverse trig forms for calculus. I would not have passed Calc2 without it.
http://www.myquizcards.com
Hi there, can someone give me some help with this question please:
The equation x^4+40x+39 has 4 roots. if two of the roots are complex conjugate roots 2+J3 adn 2-J3 by a process of long division and solving a quadratic equation find the other two roots.
Now I took the complex roots and did the following x-2+J3=0 and x-2-J3=0 therefore (x-2)^2 – (J3)^2 = x^2-4X+4-J9^2 which equals x^2-4x+13. (I hope I got this right!)
I then took the original equation adn using the quadratic I had gotten from myroots i tried to do the long division, however i always seem to end up with a remainder, I am right in think that tis should divide evenly if I have roots?
Any help on this aprecciated!
Ally
Math idiot trying to figure rate of prison-admission rate for blacks and whites in my county as a percentage of the population of each group. We have a total population of 12,785, about 2,900 of whom are black and 9,670 are white. In one year, 21 whites and 11 blacks went to prison for a total of 32; another year, the figures were 13 white and 31 blacks. How do I come up with a meaningful comparison along the lines of what I said above????????? email me with answer at tony(AT)pamlicoink.com
Hi Ally,
I’m sorry for the long delay in reply, but since your question sounded like a homework question and since I’m not really updating the blog very often at the moment, I let it slip.
I don’t follow your “therefore” statement. The point is that you have a complex conjugate pair of roots, call them x1 and x2. Then (x-x1) and (x-x2) are factors of the quartic, so the quadratic (x-x1)(x-x2) is also a factor, and this is the quadratic you derived, namelt x^2-4x+13.
If you divide the quartic by the quadratic using long division, you do indeed get a remainder of 0, the other quadratic being x^2+4x+3=(x+1)(x+3).
Tony, I would suggest the following, assuming that population levels are stable. If 32 of 12,785 people go to jail, then that’s (32*100)/12,785=0.25% – quarter of one per cent of the total population were sent to jail. The following year this figure was 0.34% – assuming that the population hasn’t changed, which it surely has. Indeed, if population rose then this seeming increase might be cancelled out.
In terms of comparison, the percentage of the white population sent to jail in the first year was (21*100)/9670=0.22%, while in the same year the percentage of the black population sent to jail was (11*100)/2900=0.38%. That’s a huge difference already. Specifically, the rate of incarceration in the black population is 173% of that of the white population – 100*(0.38-0.22)/0.22=73% extra, so the total rate is 100%+73%.
If the population levels and the ethnic mix is the same in the other year (if not, you must change the figures otherwise they have no meaning), then the situation is far worse for the black community: 0.13% of whites and 1.07% of blacks were jailed. That is, blacks were jailed at a rate of 723% of whites. Staggering.
I am Amanda and I just found this site last night, what a good domain for forum – reallyhardsums.wordpress.com
I think that stay here for a long time
Amanda
Is this gonna end someday??
Dear Sir,
I have solved the series for approximate solution (curve fitting, vague solution) but unable to find some closed form. Having attended couple of your workshop I believe you could through some light on it. Please give some reading or hint to solve it. This looks some combination not straight arithmetical, geometric or harmonic form. Might be some combination/approximation of exponential variable. Please let me know if further elaboration is needed. These number are experimental, obtained finding number of cycles from root-root in given order of Markov chain.
Problem: to find closed form function/ equation that represents the following series.
Series:
Input:
n = 1, 2, 3, 4, 5, 6, . . . . . . . . . . . . . . . .. .
N = 2^n= 2, 4, 8, 16, 32, 64, . . . . . .
Output:
K = 1, 2, 7, 74, 13610, 562523732, . . . . .
How to predict future values!!
Thanks in advance
Khurram Shahzad
SEECS-NUST
Hi Khurram,
Nice problem! Well, finding a closed-form solution can be more a matter of luck and repeated attempts than skill. I suggest you keep trying, and read a book like Abramowitz & Stegun’s “Handbook of Mathematical Functions”, and maybe also try the Online Encyclopedia of Integer Sequences for a close match.
Hi,
I’ve just started a course with the OU (UK) on Calculus of Variations and was hunting around for background info when I came across your peice on ‘Euler’. Very interesting, I’d never thought that something (photons) would need to know where to go to get there in the shortest time. But what is the difference between being constrained to a path which forces the something to take the shortest path and having to know before hand how to get there along the shortest path. Wouldn’t the former just mean we haven’t found out what ‘forces’ the photon along that path rather than say it needs to know how to do it beforehand.
PS My background is practical engineering and proofs are anything but beautiful. I just want something that work.
Hi Brian. The blog is dead, long live the comments page!
The idea of taking the shortest path is not that the photon somehow calculates the entire path, but that the “least action principle” operates over infinitessimally small distances, the distances of the calculus. So work is minimised over tiny distances, and this leads to the shortest path idea. Quite why our universe should obey such a rule is beyond me.
Hi, I’m working on a derivation for a geochemical system (describing chemical equilibrium) and I come across the following problem.
I have an equation (representing a simple kinetic system of forward and backwards reactions):
dCa/dt = K1 – K2*Ca
where K1 and K2 are constants. I can integrate this equation by separating the variables, and assuming that Ca at t=0 is 0. Doing this I get:
Ca = K1/K2 * (1-e^(-K2*t))
However, I want to extend the mathematics to use arbitrary time-dependent functions f(t) and g(t) such that:
dCa/dt = f(t)-g(t)*Ca
I now can’t integrate by separating variables. Can you tell me if it is possible to formulate an expression for Ca purely in terms of t, f(t), g(t), F(t) and G(t) (where F(t) is the integral of f(t), and G(t) is the integral of g(t))? Using integration by parts just leaves me an expression that is littered with dCa/dt terms which doesn’t help me a bit!
Thank you very much!
Cheers,
Dan
Hi Dan, and thanks for your post.
The tool you need is known as an integrating factor – I’m sure Wikipedia can help. Essentially, you use the integrating factor to turn the left hand side of your equation into an exact derivative, and then integrate both sides. Here, the integrating factor is e^G, using your notation. Then the left hand side becomes d(Ca*e^G)/dt =f*G and you go from there. Unfortunately, for unknown f and g, you won’t find a closed form solution without integral signs in it. Hope this helps.
Thank you very much Phil! I’ll have a play around with this and see what I can make of it. I don’t mind not having a closed form, provided I have an equation with Ca only on the LHS. The arbitrary time-dependent functions can be manipulated as algebraic expressions and – if my hunch is correct – will eventually cancel from the equation(s) during the rest of the derivation.
I appreciate your help!
Cheers – Dan
OK. Another question relating to the stuff I’m working on:
I have two equations:
Equation1 = (integral sign) A(t)*b(t)*dt + C1
C1 is a constant, and evaluated such that Equation1 = 0 when t = 0
Equation2 = (integral sign) A(t)*dt + C2
C2 is a constant and evaluated such that Equation2 = 0 when t=0
Even though A(t) and b(t) are arbitrary functions, is there anything definite we can say about the _ratio_ of Equation1 / Equation2 as t -> 0?
How about ln(Equation1 / Equation2) as t -> 0?
How about d/dt [ln(Equation1 / Equation2)] as t-> 0?
It strikes me that because the same function A(t) appears in both equations, the behaviour of the system _may_ be mathematically constrained.
If you want more detail:
A(t) = [ f(t)*e^ (G(t)) ] and
b(t) = [ e^ ((k-1)*G(t)) ]
where f(t) and G(t) are different arbitrary functions, and k is a constant.
Thank you again! I hope I’m not abusing the spirit of this blog by bombarding you with this stuff.
Not sure off the top of my head, Dan. You might want to look into some of the theory about perturbation methods, and asymptotic analysis. These enable you to estimate rough sizes of functions in some limit (they do more than that, but this is what is relevant to you), and this might help if the functions you mention are of a suitable sort. Sorry I can’t be more specific or helpful at the moment.
Sorry – to clarify:
A(t) = f(t) * e^(G(t))
b(t) = e^((k-1)*G(t))
I forgot that square brackets have a specific mathematical meaning.
Thanks once again Phil. Looks like I’m moving into pretty convoluted territory here. I appreciate your help.
can anybody help me, i am trying to get the stability region of a numerical scheme, which i derived an equation in which when i plot the graph i will be able to determine the stability region.
Can anybody explain how i could plot the graph of:
h(theta)=exp(i*theta) – 1 on a cartesian plane.
Pls help me
Hi Ehigie. As you probably know, as theta varies from 0 to 2pi, exp(i*theta) traces out a circle of radius 1, centred on the origin. h(theta) is the same circle, but shifted to the left by 1 (think about the real and imaginary parts of exp(i*theta)).