Eyesight = a (1 – r) ^ x

Eyeball

The Gemara quotes a statement of our Sages that when a person takes a very large step he loses 1/500th of his eyesight.   Tosfos points out the issue that based on this a person who takes 500 large steps would be blind, yet we see people who do this and can still see. This is based on the assumption that each step reduces eyesight by 1/500th of the persons starting eyesight. Tosfos explains that this is not the case. Each step only takes away 1/500 of the eyesight that they have at that moment. How much of his eyesight would be left after taking 500 large steps? (Taanis 10b)

 

 

The formula for exponential decay is

y = a (1 – r)x

y = current eyesight

a = original amount of eyesight (in this case 100%)

r= rate of decay (in this case 1/500)

x=number of steps (in this question 500)

 

y= 100 (1- 1/500)500

y= 100 (1-.002)500

y= 100 (.998)500

y= 100 * .3675112548

y = 36.75112548% of vision remaining

(229)