Math Is Fun Forum

Hi!

I think there's only 1 solution, and I wrote it down, z = -W(-1)

Right?

Are you sure about the distances of the antinode to the sources?

I don't know exactly what does 2nd order antinode mean. If it would be the 1st antinode in the 2nd harmonic of the standing wave, the following should be true:

And it doesn't make sense.

If the 2nd order antinode means the 2nd antinode for a certain harmonic [n], then the following holds:

Or

And neither of both makes sense, since [n] should be an integer.

Hi!

Lets see if this works. We have the following relation between the wave speed [c] (m/s), the wavelength [lambda] (m) and the frequency [f] (Hz):

Then, for air you use c = 344 (m/s):

Now, for steel you can calculate:

We now have all the information to answer the time it takes to travel s = 84 (m):
(a) For air

(b) For steel

Hi!

I am sure there will be an easier way to solve it, but I will try to say how would I do it.

I am going to assume you have an original point in the XY plane, heading north, which I will consider in the negative part of the X axis. Then, the original coordinates would be (-x, 0, 0), being the distance to the origin of coordinates equal to x.

I also consider that rotation from north and clockwise is keep the point in the XY plane, so rotate in Z an angle q1 = -33. And then doing the same for a "dip"angle, I assume it is rotate with respect to X, an angle q2.

First, you have to calculate the new coordinates of the point, for example from http://www.siggraph.org/education/materials/HyperGraph/modeling/mod_tran/3drota.htm

1. Rotation in Z:

2. Likewise, rotation in X:

Yo can actually find the 2 matrices, multiply one with the other and obtain the total displacement.

Now have point (x'', y'', z''). Then, we have to move it in the direction normal to certain plane, a distance of 50. Ok, the equation characterizing a plane using a vector normal to it, (a, b, c) that passes through point (x0, y0, z0) is:

And the line passing through (x'', y'', z'') normal to the plane can be written using an independent variable t:

In total, you have to calculate what new point (x''', y''', z''') is at distance 50 from (x'', y'', z''):

So the point must satisfy the equation of the line and of the distance.

And that's what I can say with the info you provided

Jose

Haha, you're welcome!

It is curious that you calculated the area underneath f(y) correctly, and not for f(x). That for f(y) is much more difficult!

Jose

Hi!

You can use Taylor series approximation around a = 1 for example, or any other a closer to 2:

Then, at x = 2

The result given is using up to 1/10. It is still not very precise, since ln(2) = 0.6931, but you can keep on adding and substracting fractions to get it more accurate.

I guess that if you approximate it closer to 2, for example a = 1.9, it will converge faster, but then you have to use powers in your calculator (or do many more multiplications )

Actually, I think that approximating around 1 is the only solution to only use (+, -, ×, ÷), because if doing it around a = 1.9, you need to know f(1.9) = ln(1.9). However, with a = 1, I guess ln(1) = 0 is well known

Jose

Hi!

You have to differentiate in order to obtain the pdf out of the cdf!

I think then they results match!

Jose

Hi!

If you would like to understand HOW to calculate f(y), from Wikipedia [http://en.wikipedia.org/wiki/Probability_density_function]:

The probability contained in a differential area must be invariant under change of variables y = g(x)

This was for the simple case of a monotonic function g. There is a general expression in Wikipedia as well.

If you would like to know WHAT does it mean, I can only provide an example: Random Number Generation. Rather than explaining it myself, I think this other article will be useful [http://en.wikipedia.org/wiki/Inverse_transform_sampling]

Jose

Hi!

I guess you need the steps, but I cannot figure them out now

Sometimes, the result can give some clue, even though it's not the case for me now. Anyway, Mathematica says:

That's a weird result, though! Hope you get luckier than me

Jose

Hi

The first one I would say:

So every requirement holds.

The second:

Hope it helps!

Jose

Hi!

I think in this website they explain congruences very well http://primes.utm.edu/glossary/xpage/Congruence.html

Suppose a, b and m are any integers with m not zero, then we say a is congruent to b modulo m if m divides a-b. We write this as

    a = b (mod m).

For your example, use the transitive property: If a = b (mod m) and b = c (mod m), then a = c (mod m).

Then, lambda = 0.25 (mod 23) and 0.25 = 6 (mod 23), then lambda = 6 (mod 23). Where the second is true because 23 / (0.25 - 6) = -4

Jose

Hi!

Apparently, according to http://en.wikipedia.org/wiki/Sum

Moreover:

where Bk is the kth Berboulli number.

And in http://en.wikipedia.org/wiki/Bernoulli_number

Jose