exponential/compounded growth/decay

**Såxon** · February 28th 2009, 03:16 PM

So confusing,
So I seem to be having a lot of trouble with putting compounded growth/decay related problems into context with the formulas.
The problem is:
A pie is removed from a 375 degree Fahrenheit oven and cools to 215 degrees. after 15 minutes in a room at 72 degrees fahrenheit. How long from the time it is removed from the oven will it take the pie to cool to 120 degrees fahrenheit?
Im really not sure where to start but here goes
A = P(1+r/k)^kt
120 = 375(1+15/160)^(160t) now just use log to solve???
did i set this up correctly?

**e^(i*pi)** · February 28th 2009, 03:48 PM

Originally Posted by Såxon

So confusing,
So I seem to be having a lot of trouble with putting compounded growth/decay related problems into context with the formulas.
The problem is:
A pie is removed from a 375 degree Fahrenheit oven and cools to 215 degrees. after 15 minutes in a room at 72 degrees fahrenheit. How long from the time it is removed from the oven will it take the pie to cool to 120 degrees fahrenheit?
Im really not sure where to start but here goes
A = P(1+r/k)^kt
120 = 375(1+15/160)^(160t) now just use log to solve???
did i set this up correctly?

Use Newton's Law of Cooling:

$\frac{dT}{dt} = -kT$

which when integrated gives:

$T = T_{0}e^{-kt}$ where $T$ is final temperature, $T_{0}$ is initial temperature, $k$ is a constant and $t$ is time

Plug in what you know:
$<br /> 215 = 375e^(-15k)$

$<br /> k = -\frac{1}{15}ln{\frac{215}{375}}$

Now we know k we can use it to find t:

$t = -\frac{1}{k}ln({\frac{T}{T_{0}})} = -\frac{1}{\frac{1}{15}ln{\frac{215}{375}}}ln({\frac {120}{375})}$

**Såxon** · February 28th 2009, 04:16 PM

Originally Posted by e^(i*pi)

Use Newton's Law of Cooling:

$\frac{dT}{dt} = -kT$

which when integrated gives:

$T = T_{0}e^{-kt}$ where $T$ is final temperature, $T_{0}$ is initial temperature, $k$ is a constant and $t$ is time

Plug in what you know:
$<br /> 215 = 375e^(-15k)$

$<br /> k = -\frac{1}{15}ln{\frac{215}{375}}$

Now we know k we can use it to find t:

$t = -\frac{1}{k}ln({\frac{T}{T_{0}})} = -\frac{1}{\frac{1}{15}ln{\frac{215}{375}}}ln({\frac {120}{375})}$

I came up with a negative number, and the answer is in minutes?? i'm confused, how can this be the answer?

**e^(i*pi)** · February 28th 2009, 04:31 PM

Originally Posted by Såxon

I came up with a negative number, and the answer is in minutes?? i'm confused, how can this be the answer?

Ah right, I forgot about the sign on k in the final equation. That front minus sign needn't be there[/color]

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