/sci/ - Science & Math

>>11563159
>>11563199
You can pretty easily find the big circle to have radius [math] R_0=1-\sqrt2/2 [/math]. Then, the line joining the center of the big circle to an acute corner is [math] m=\sqrt{R_0^2+(\sqrt{2}/2)^2}=\sqrt{2-\sqrt{2}} [/math] (which is different from what you had, however I verified all this in solidworks). Noticing this line runs along all the circles' centers, you know that [math] R_0+2R_0a+2R_0a^2+...=m [/math]. So [eqn] 2R_0\sum_{n=1}^{\infty}a^n=m-R_0\implies\sum_{n=1}^\infty a^n=\frac{a}{1-a}=\frac{m-R_0}{2R_0}\implies a=\frac{\frac{\sqrt{2-\sqrt{2}}}{2-\sqrt{2}}-\frac{1}{2}}{\frac{\sqrt{2-\sqrt{2}}}{2-\sqrt{2}}+\frac{1}{2}} [/eqn]
which I really really cannot be arsed to simplify but again was verified in solidworks. Since [math] a [/math] is the ratio of consecutive radii, it now shouldn't be too hard to find the area you are after. [eqn] A=\pi R_0^2 +2\sum_{n=1}^\infty\pi(a^nR_0)^2=\pi R_0^2+2\pi R_0^2\bigg[\frac{a^2}{1-a^2}\bigg]\approx 0.404 [/eqn] i think. this most likely could have been done a LOT cleaner.

>>11564269
Say you are determining hardness of a material, and you have a very large sample size. The way I understand it, is if there are multiple factors all contributing to hardness, and if they are all randomly determined, and if they are all equally likely to influence hardness, then you get a normal distribution of hardness for the population. And of course, if you took multiple readings of each sample, the distribution of the mean hardness of each specimen tends to a normal distribution over the population per the central limit theorem. Other distributions come in different contexts, like the Boltzmann distribution just naturally pops up when you taylor expand the number of microstates and divide by the canonical partition function.

>>11365129
Well there ya go, you are probably taking measurements from within the entrance length (there are multiple empirical formulas out there to calculate just how long the entrance length is). Trust your velocity measurements from the pitot tubes and do whatever you need to do for the write up.
>>11365057
D'Almbert's paradox.

Out of morbid curiousity, what are all the things you can do to make a shitty baby?
The ones I know are:
>Drinking while pregnant: Causes Fetal Alcohol Syndrome
>Thalidomide: Cures morning sickness, also causes severe birth defects if taken very early in pregnancy, the most obvious defect being malformed/missing limbs