Saturday, April 25, 2009

The sound of an explosion in space

It has been said that in space, no one can hear you scream. Why? Because sound travels through particles and particles in space are few and far between. So sound doesn't usually get very far in space.

But with enough energy peculiar things happen. A black hole has been observed to sing a B flat 57 octaves below middle C in the Perseus cluster of galaxies located 250 million light years from Earth.

Would this be the only form of sound traveling in space? Doubtful... My own hypothesis is that explosions would generate a sound even in a perfect vacuum with no particles. I doubt anyone has ever done it. But I believe the particles of the explosion itself would carry kinetic energy until colliding with something... anything that could carry sound... such as the hull of a space craft. Now I just need someone to go up there and test it.

Another question: what is the speed of sound in space? If a tree exploded in outer space and no one was there? Would that make a sound?

Clock angle

It is 6:30 on a normal clock with a face. What is the angle between the hour hand and the minute hand?

What would the angle be at 6:35.27?

Sunday, March 29, 2009

Special Right Triangles

There are the only 2 angle based special triangles. 45 45 90 is special because it is isosceles as well as right. Using the Pythagorean theorem and the fact that it has two equal sides, we can find out the ratio of all sides.

Example: If the legs are length n, then the hypotenuse will be
sqrt(n^2 + n^2) <------ by the Pythagorean theorem. sqrt(2n^2) n * sqrt(2)

30 60 90 is special because it is a equilateral triangle cut in half. Therefore one of its sides is half the length of another (the hypotenuse) The third side can be found with the Pythagorean theorem.

An equilateral triangle has a length 2n. If we draw an altitude, it will bisect the triangle and create 2 congruent right triangles. The bases of the right triangles are n. The altitude is
sqrt((2n)^2 - n^2)
sqrt(4n^2 - n^2)
n * sqrt(3)

It is also possible to find more "special" angle triangles whose angles are 15 75 90 with the angle addition formulas.

There are plenty more special right triangles whose nature is based on sides rather than angles, such as the 3-4-5 and 5-12-13. There is an infinite number of such triangles and they can be found easily.

It has already been proven that the difference between squares of numbers n^2 and (n+1)^2 is an odd number 2n+1. Because of this, all odd numbers are the difference of some pair of squares n^2 and (n+1)^2.

Choose any odd number and use it as the side of a right triangle. Because it is odd, its square is also odd. Because its square is odd, its square is the difference between a pair of squares. So if the remaining sides are n and n+1, then our chosen side will be sqrt(2n+1). Square our chosen odd number, subtract 1 from it and divide by 2 to get n, which is the length of the other leg. The final side is the hypotenuse and is n+1.

For example, let's try 21.

21 = sqrt(2n+1)
441 = 2n+1
220 = n
221 = n+1

21, 220, and 221 form a Pythagorean triple.

21^2 + 220^2 = 221^2
441 + 48400 = 48841 <----- TRUE!

Sunday, March 15, 2009

Send me your math problems!

If you need help understanding a math concept in algebra, calculus, trig or even basic math, send me your questions. I'll show you how to solve the problem and if it's interesting, I might put it on the blog as an example.

Friday, March 13, 2009

Reasoning Challenge

How many times does the earth rotate on its axis in one year?

Wednesday, March 11, 2009

Welcome to Math ACE!

The first math challenge is to prove the following trig identity:

Submit your answers via comment.