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} \ $$