anonymous
  • anonymous
Obtain a result for calculating the time required to consume a finite resource. dM/dt=-Pexp(at)+Fexp(-ut)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
Whare are F, P, and M?
anonymous
  • anonymous
M is the current size of resources,Pexp(at) is rate of production of new resource and Fexp(-ut) is the rate new resource deposits are being discovered
anonymous
  • anonymous
Basically supposed to solve for the time t

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anonymous
  • anonymous
You need to solve M(t) = 0.
anonymous
  • anonymous
So you'd have to integrate first.
anonymous
  • anonymous
this is where i'm stuck at. M=(-1/a)Pexp(at) +P/a +F/a _(1/a)Fexp(-ut)
anonymous
  • anonymous
Okay, what is the anti derivative of \(\exp(at)\)
anonymous
  • anonymous
(1/a)exp(at)
anonymous
  • anonymous
Where is the constant of integration?
anonymous
  • anonymous
sorry. was integrated from t=0 to t=t
anonymous
  • anonymous
Alright so what are you getting \(M(t)\)?
anonymous
  • anonymous
M=(-1/a)Pexp(at) +P/a +F/a _(1/a)Fexp(-ut)
anonymous
  • anonymous
How are you getting that though?
anonymous
  • anonymous
the original equation was dM/dt=-Pexp(at)+Fexp(-ut)
anonymous
  • anonymous
Where is +P/a coming from though?
anonymous
  • anonymous
after the limits were put in
anonymous
  • anonymous
from 0 to t
anonymous
  • anonymous
I mean F/a
anonymous
  • anonymous
Sorry. That should be F/u
anonymous
  • anonymous
M=(-1/a)Pexp(at) +P/a +F/u -(1/u)Fexp(-ut)
anonymous
  • anonymous
Now you would find the roots.
anonymous
  • anonymous
Roots of which term?
anonymous
  • anonymous
roots of t
anonymous
  • anonymous
as in an equation of the exp(at) and exp(-ut) terms?
anonymous
  • anonymous
Basically M = 0, and you have to find t values where that expression is 0
anonymous
  • anonymous
The trivial case is t=0
anonymous
  • anonymous
\[ 0=\frac Pa (1-e^{at}) + \frac Fu(1-e^{-ut}) \]
anonymous
  • anonymous
\[ \frac Pa (1-e^{at}) = - \frac Fu(1-e^{-ut}) \]
anonymous
  • anonymous
Hmmmm\[ -\frac{Pu}{Fa}(1-e^{at}) = 1-e^{-ut} \]
anonymous
  • anonymous
This is the question. Obtain an analytic results that can be used for calculating the time required to consume a finite fuel resource. Currently the resource is of size, Mo, which is being produced at a rate ,P. The rate of consumption is growing exponentially at a relative rate ,Pexp(at), and new fuel resource deposits are being discovered at a rate, Fexp(-ut)
anonymous
  • anonymous
That changes things a bit.
anonymous
  • anonymous
So I formulated this equation dM/dt=-Pexp(at)+Fexp(-ut)
anonymous
  • anonymous
Well, if you integrate from 0 to t you will \(\Delta M\) rather than just \(M\).
anonymous
  • anonymous
\[ \Delta M = M-M_0 \]
anonymous
  • anonymous
I think to find the roots in this case, we'll have to use the natural log.
anonymous
  • anonymous
ln of -Pu/Fa=1-exp(-ut)/1-expt(at) ?
anonymous
  • anonymous
Yeah, but we'd have to factor in the \(M_0\) as well.
anonymous
  • anonymous
This problem is obnoxious.
anonymous
  • anonymous
\[ M(t) = M_0 +\int_0^t\frac{dM}{dt}dt \]
anonymous
  • anonymous
Since we want \(M(t)=0\), we have: \[ -M_0 = \int_0^t\frac{dM}{dt}dt \]
anonymous
  • anonymous
is that a possibility?
anonymous
  • anonymous
Yes, it's possible.

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