Math Forum
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!
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Forgive that beside You was little ed!
Excellent forum, added to favorites!
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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!