Let's discuss Mathematics

I'm not sure if there's any useful insight. I've just come across with these two questions, but never gave them thought myself. So, to force myself to do that I posted them here.

I'm not entirely sure how you'd even do the latter with Excel. Take a very fine grid and calculate?

 

Just a whole bunch of simulations with pairs of coordinates. See where the average tends to...

 

For fun's sake I did the same with python:

Spoiler :

Code:

from math import sqrt
from random import random

def d(x,y):
  dx = (x[0] - y[0])**2
  dy = (x[1] - y[1])**2
  return sqrt(dx+dy)

def randomDist(pointCount):
  sum = 0.0
  for i in range(0,pointCount):
    x = [random(),random()]	
    y = [random(),random()]	
    sum += d(x,y)
  return ( sum / pointCount )

# MAIN:

print(randomDist(1000000))

A few runs puts it in the interval [0.521, 0.522]. Gonna think about how to do it properly later.

 

1) Based on mathematical intuition, I believe it doesn't depend on \alpha. (If you had a quarter circle and had to divide it in half, you could just divide the full circle in an inner and outer half, and then cut it into 4 pieces later).
Assuming you make a 'round' cut at constant radius, it has to be at 1/2 Sqrt(2) of the radius of the full Brie cheese.

 

I've managed to get the answer to the second problem in the form of an integral (a horrible looking integral but an integral all the same). Now I just need to integrate it. Will get back to you as soon as I've found the answer.

 

I was thinking you'd make one straight cut, not a circular one.

As for puzzle two, I'm hoping there would be some easier way than the straight up integration of distances.

 

In that case I get x = sqrt(alpha/(2*sin(alpha))), where x is the distance from the tip (as a fraction of the radius) at both edges for the two points that define the cut and alpha is in radians.

For alpha << 1 this reduces to dutchfire's answer.

 

That answer fails at larger alpha. For example, at alpha > 0.6pi, x>1.

Re: 2) It looks hard to calculate it exactly, the integral is very ugly. It is much easier if you're doing this on a periodically continued square, because then there are only two variables effectively and the answer is 0.38. This is a strict lower bound for your question (although obviously not a very good one).

The question also looks much more tractable if you use another norm, for example the max-norm or the taxi-cab metric l_1 metric. Maybe that way it is possible to derive some other bounds on the answer.

 

That's what I got too.

It's valid at least for cases alpha < pi/2, which the most brie pieces conform to. If alpha goes near pi, it's impossible to cut it in half from straight side to straight side, as the triangle becomes extremely small.

The problem #2 is from YouTube, which suggested a video posing this question. I haven't watched it yet (to not spoil the fun), but it could be something more accessible. If you use different norms, perhaps Hölder's inequality could give at least some estimate.

 

Problem 2 is ill-defined. The answer depends on the distribution you choose for the points. Even "natural" choices may lead to different answers. See

https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

 

I don't see how. Here's how I see the problem:

Points = real number pairs in [0,1]x[0,1].
Distance = the usual euclidic distance.
Average of an integrable function over set A is
1/|A| * \int_A f dm, where m = Lebesgue measure.

Thus, the average of the distance is
\int\int\int\int sqrt{ (x_1 - x_2)^2 + (x_3 - x_4)^2 } dx_1 dx_2 dx_3 dx_4.

I suppose you thought that it's the expected value of the distance of two random points?

 

Try this - For a project in math class, Anne is making a cover for a circular lounge chair 3 feet in diameter. The finished lounge chair needs to hang down 5 inches over the edge of the chair all the way around. To finish the edge of the cover, Anne will fold under and sew down 1 inch of the material all around the edge. Anne is going to use a single piece of rectangle fabric that is 60 inches wide. What is the shortest length of fabric, in inches, Anne could use to make the cover without needing to attached several pieces of fabric together?

Spoiler :

48 inches - Before doing anything else, make sure you convert all your measurements into the same scale. Because we are working mainly with inches, convert the table with a 3 foot diameter into a table with a (3)(12)=(36) inch diameter.

Now we know that the tablecloth must hang an additional 5+1 inches on EVERY side, so our full length of the tablecloth, in any straight line, will be:

1+5+36+5+1

48 inches.

 

I just wanna know how putting this cover over it turned the lounge chair into a table.

 

feet and inches

I'll try to look at it later

 

Gotta be honest, I can't parse through the english in that problem

 

I thought this thread was supposed to be about mathematics and not conversions between silly units.

 

you convert by means of mathematic

 

QUESTION: Show that number n^3 - 3n^2 + 2n is divisible by 3. n is integer.

I can't solve it. I tried sth like: the polynome = 3x, x is integer. But not sure how to show what I need by that... help?

 

Assume that it isn't divisible by 3 and try to prove by contradiction. Not sure if that'll work but that's what I would try first.

 

Is it:

n^3 - 3n^2 + 2n = n (n - 1) (n - 2)

One of n (n - 1) (n - 2) will always be a multiple of 3.

Spoiler :

How to factorize n^3 - 3n^2 + 2n? Put it into Wolfram Alpha.