Mathematical formula for a pattern

Question

I was cooking and saw a chart of temperature vs cooking time like so:

$154\mathrm{F}$ - $36$ hours
$164\mathrm{F}$ - $18$ hours
$174\mathrm{F}$ - $9$ hours

Basically for every $10\mathrm{F}$ increase in temperature, cooking time is halved. Now I'm thinking, there must be a formula to easily figure out if temperature is $x$, then cooking time is $y$. Or vice-versa, if I want to cook it for $y$, then I should set temperature to $x$.

I was racking my brain trying to figure this out. I feel like it should be obvious :(

@Stefan: I disagree with your edit. Unit symbols such as kg, N, and °F are never set in italics. — , Mar 01 '13 at 08:04
@RahulNarain: You're absolutely right. I re-edited it - thanks. — Stefan Hansen, Mar 01 '13 at 08:15

score 7 · Accepted Answer · answered Mar 01 '13 at 07:22

7

$$x = 154 +10n \implies y = \dfrac{36}{2^n}$$ Hence, $$y = \dfrac{36}{2^{(x-154)/10}}$$

answered Mar 01 '13 at 07:22

How elegant! Thank you! – pixelfreak Mar 01 '13 at 07:27
1

@pixelfreak: And of course $x=154+10\log_2(36/y)$. – Mar 01 '13 at 07:52

Mathematical formula for a pattern

1 Answers1