User:Plutor/Math sandbox

I use this page to create LaTeX math images for random things. See Help:Formula for documentation.

$\sigma _{0}=\langle \sigma _{m}\rangle \approx \int _{0}^{\varepsilon _{0}}{\rm {d}}\varepsilon _{0}\left\lbrace 2\kappa {\frac {n(\varepsilon _{0})-n(\infty )}{n(0)-n(\infty )}}+\alpha \left[v_{p}(\varepsilon _{0})-v_{p}^{\max(}\varepsilon _{0})\right]\right\rbrace$

Carrying change

2.7 miles = 1.35 hours
1 lb = 453.59237 g
penny = 2.5g = 250 g/$
      = 0.551155655 lb/$ * 1.35 cal/lb*day
      = 0.74406013425 cal/$*day

Value-density of a penny:

${\begin{matrix}\mathbb {D} &=&{\frac {2.5g}{\$0.01}}\\\ &=&250\;g/\$\\\ &\approx &0.551\;lb/\$\end{matrix}}$

Energy used carrying a penny around for a day:

${\begin{matrix}\mathbb {E} &=&\mathbb {D} *{\frac {2.7\;mi}{2\;mph}}*{\frac {1\;cal}{lb*hr}}\\\ &=&\mathbb {D} *1.35\;cal/lb\\\ &\approx &0.744\;cal/\$\end{matrix}}$

When is Powerball worth it?

For http://plutor.org/blog/2006/02/15/when-is-powerball-worth-it/

$\ {\begin{matrix}v_{ticket}&>&\$1.00\\\\v_{ticket}&=&{\frac {v_{jackpot}}{p_{jackpot}}}+{\frac {v_{2}}{p_{2}}}+{\frac {v_{3}}{p_{3}}}+\cdots +{\frac {v_{n}}{p_{n}}}\\\\\$1.00&<&{\frac {v_{jackpot}}{146,107,962.00}}+{\frac {\$200,000}{3,563,608.83}}+{\frac {\$10,000}{584,431.85}}+{\frac {\$100}{14,254.44}}+\\&&{\frac {\$100}{11,927.18}}+{\frac {\$7}{290.91}}+{\frac {\$7}{745.45}}+{\frac {\$4}{126.88}}+{\frac {\$3}{68.96}}\\\$1.00&<&{\frac {v_{jackpot}}{146,107,962.00}}+\sim 0.19711512\\\\\$117,307,873&<&v_{jackpot}\end{matrix}}\$