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This was a question in math league a while ago, and I did not know how to do the problem.
Givens: There are 3 semi circles, one with radius 3, one with radius 2, and one with radius 1. They are inlaid as shown in the diagram. There is a circle tangent to the sides of each of the semicircles. What is , the radius of that circle?
What I attempted to do before getting confused: I made a triangle (not drawn in the diagram), connecting the center of the cicle with the center of the semicircle radius=1, connecting the circle center with the center of semicircle radius=2, and the final line connecting the two centers of the semicircles r=1 and r=2. So now I got a triangle with sides of 3, 1+x, and 2+x.
Was making a triangle helpful in solving the problem? What is the next step I should take? Spencer T♦ C 01:36, 27 August 2010 (UTC)
Break the side of length 3 into two parts: from the center of the circle of radius 2 to the center of the circle of radius 3, of length 1, and from there to the center of the circle of radius 1, of length 2. Draw a line from the center of the big circle (of radius 3) to the center of the circle of unknown radius, and keep going until that line hits the point where the big circle and the unkown circle touch each other. (Notice that if a line passes through the centers of two circles in the same plane that touch each other, then it must pass through the point where they touch each other.) The portion of that line that lies within your triangle has length 3 − x. Keep going from there. Michael Hardy ( talk) 02:01, 27 August 2010 (UTC)
What is the oldest open problem in mathematics? -- 84.61.172.89 ( talk) 07:53, 27 August 2010 (UTC)
Although ln(z) is multivalued and discontinuous near 0, i'm thinking sin(ln(z)) collapses the multivalues into a single valued function and is continuous. Am i correct? If so, what's the Laurent series? Thanks, 24.7.28.186 ( talk) 09:57, 27 August 2010 (UTC)
Does have multiple solutions for when is not an integer, and if so, how many?-- Alphador ( talk) 11:23, 27 August 2010 (UTC)
I have the following definition in my book:
Definition: If a poset T has a smallest element 0, then any cover of 0 is called an atom or point of T. A poset with 0 is atomic if every nonzero element contains an atom.
Now can someone give me an example of a non-atomic poset. Since 0 is the smallest element all nonzero elements contain it, and so either are atoms or contain atoms. So isnt any poset with 0 an atomic poset?- Shahab ( talk) 11:46, 27 August 2010 (UTC)
find the wxtreme values of function —Preceding unsigned comment added by Himanshu.napster ( talk • contribs) 19:23, 27 August 2010 (UTC)
For instance, we can write and thus and . (Where fx and fy denote the partial derivatives of f with respect to x and y, respectively.) Determine the ordered pairs (x,y) for which both fx(x,y) and fy(x,y) vanish. Once you have done this, compute the determinant of the 2x2 Hessian matrix of f:
(The above partial derivatives are all "second order" partial derivatives of f.) Then, evaulate this determinant at each of the critical points to test whether they are extrema or not. I will leave you to work out how to do this as an exercise. PS T 06:31, 28 August 2010 (UTC)
There is a calc problem that I don't understand at Wikipedia:Reference desk/Science#What is the wind-water-solar climate change mitigation scenario atmospheric carbon projection? Why Other ( talk) 22:22, 27 August 2010 (UTC)
How can I get the moment generating function of the square of a standard normal random variable? Do I just replace with in the integral? I know it's meant to come out the same as a gamma mgf but I can't get it to work. —Preceding unsigned comment added by 118.208.51.232 ( talk) 23:08, 27 August 2010 (UTC)
Let me try doing this...
Alas! I'm sure the answer is supposed to be . What am I doing wrong? —Preceding unsigned comment added by 130.102.158.15 ( talk) 06:02, 28 August 2010 (UTC)
Wrong!
is not the same as
You were OK until that point.
More later.... Michael Hardy ( talk) 18:50, 28 August 2010 (UTC)
I see Meni Rosenfeld already noted the error.
Now remember that once you've got
then via the substitution of ax for x you get
and hence
provided a > 0. (If a is negative, then the bounds of integration get reversed, and you go on from there....)
Now apply this in the case where
Michael Hardy ( talk) 19:00, 28 August 2010 (UTC)
Mathematics desk | ||
---|---|---|
< August 26 | << Jul | August | Sep >> | August 28 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
This was a question in math league a while ago, and I did not know how to do the problem.
Givens: There are 3 semi circles, one with radius 3, one with radius 2, and one with radius 1. They are inlaid as shown in the diagram. There is a circle tangent to the sides of each of the semicircles. What is , the radius of that circle?
What I attempted to do before getting confused: I made a triangle (not drawn in the diagram), connecting the center of the cicle with the center of the semicircle radius=1, connecting the circle center with the center of semicircle radius=2, and the final line connecting the two centers of the semicircles r=1 and r=2. So now I got a triangle with sides of 3, 1+x, and 2+x.
Was making a triangle helpful in solving the problem? What is the next step I should take? Spencer T♦ C 01:36, 27 August 2010 (UTC)
Break the side of length 3 into two parts: from the center of the circle of radius 2 to the center of the circle of radius 3, of length 1, and from there to the center of the circle of radius 1, of length 2. Draw a line from the center of the big circle (of radius 3) to the center of the circle of unknown radius, and keep going until that line hits the point where the big circle and the unkown circle touch each other. (Notice that if a line passes through the centers of two circles in the same plane that touch each other, then it must pass through the point where they touch each other.) The portion of that line that lies within your triangle has length 3 − x. Keep going from there. Michael Hardy ( talk) 02:01, 27 August 2010 (UTC)
What is the oldest open problem in mathematics? -- 84.61.172.89 ( talk) 07:53, 27 August 2010 (UTC)
Although ln(z) is multivalued and discontinuous near 0, i'm thinking sin(ln(z)) collapses the multivalues into a single valued function and is continuous. Am i correct? If so, what's the Laurent series? Thanks, 24.7.28.186 ( talk) 09:57, 27 August 2010 (UTC)
Does have multiple solutions for when is not an integer, and if so, how many?-- Alphador ( talk) 11:23, 27 August 2010 (UTC)
I have the following definition in my book:
Definition: If a poset T has a smallest element 0, then any cover of 0 is called an atom or point of T. A poset with 0 is atomic if every nonzero element contains an atom.
Now can someone give me an example of a non-atomic poset. Since 0 is the smallest element all nonzero elements contain it, and so either are atoms or contain atoms. So isnt any poset with 0 an atomic poset?- Shahab ( talk) 11:46, 27 August 2010 (UTC)
find the wxtreme values of function —Preceding unsigned comment added by Himanshu.napster ( talk • contribs) 19:23, 27 August 2010 (UTC)
For instance, we can write and thus and . (Where fx and fy denote the partial derivatives of f with respect to x and y, respectively.) Determine the ordered pairs (x,y) for which both fx(x,y) and fy(x,y) vanish. Once you have done this, compute the determinant of the 2x2 Hessian matrix of f:
(The above partial derivatives are all "second order" partial derivatives of f.) Then, evaulate this determinant at each of the critical points to test whether they are extrema or not. I will leave you to work out how to do this as an exercise. PS T 06:31, 28 August 2010 (UTC)
There is a calc problem that I don't understand at Wikipedia:Reference desk/Science#What is the wind-water-solar climate change mitigation scenario atmospheric carbon projection? Why Other ( talk) 22:22, 27 August 2010 (UTC)
How can I get the moment generating function of the square of a standard normal random variable? Do I just replace with in the integral? I know it's meant to come out the same as a gamma mgf but I can't get it to work. —Preceding unsigned comment added by 118.208.51.232 ( talk) 23:08, 27 August 2010 (UTC)
Let me try doing this...
Alas! I'm sure the answer is supposed to be . What am I doing wrong? —Preceding unsigned comment added by 130.102.158.15 ( talk) 06:02, 28 August 2010 (UTC)
Wrong!
is not the same as
You were OK until that point.
More later.... Michael Hardy ( talk) 18:50, 28 August 2010 (UTC)
I see Meni Rosenfeld already noted the error.
Now remember that once you've got
then via the substitution of ax for x you get
and hence
provided a > 0. (If a is negative, then the bounds of integration get reversed, and you go on from there....)
Now apply this in the case where
Michael Hardy ( talk) 19:00, 28 August 2010 (UTC)