User:Plutor/Math sandbox

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

${\sigma }_0=⟨{\sigma }_m⟩\approx \underset{0}{\overset{{\epsilon }_0}{\int }}\mathrm{d}{\epsilon }_0\{2\kappa \frac{n({\epsilon }_0)-n(\mathrm{\infty })}{n(0)-n(\mathrm{\infty })}+\alpha [v_p({\epsilon }_0)-v_p^{\max (}{\epsilon }_0)]\}$

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:

$\begin{array}{ccc}\mathbb{𝔻}& =& \frac{2.5g}{\mathrm{\$}0.01}\\ & =& 250g/\mathrm{\$}\\ & \approx & 0.551lb/\mathrm{\$}\end{array}$

Energy used carrying a penny around for a day:

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

When is Powerball worth it?

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

$\begin{array}{ccc}{v}_{ticket}& >& \mathrm{\$}1.00\\ \\ v_{ticket}& =& \frac{v_{jackpot}}{p_{jackpot}}+\frac{v_2}{p_2}+\frac{v_3}{p_3}+\cdots +\frac{v_n}{p_n}\\ \\ \mathrm{\$}1.00& <& \frac{v_{jackpot}}{146,107,962.00}+\frac{\mathrm{\$}200,000}{3,563,608.83}+\frac{\mathrm{\$}10,000}{584,431.85}+\frac{\mathrm{\$}100}{14,254.44}+\\ & & \frac{\mathrm{\$}100}{11,927.18}+\frac{\mathrm{\$}7}{290.91}+\frac{\mathrm{\$}7}{745.45}+\frac{\mathrm{\$}4}{126.88}+\frac{\mathrm{\$}3}{68.96}\\ \mathrm{\$}1.00& <& \frac{v_{jackpot}}{146,107,962.00}+\sim 0.19711512\\ \\ \mathrm{\$}117,307,873& <& v_{jackpot}\end{array}$