How Long Until a Cursed Civilization's Population Reaches Zero?

[Math Henry]

Homework Statement

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Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is:
P_n = P_n-1-√P_n-1

Homework Equations

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Considering that the civilization starts with a population of 676 people, how many week until it reaches zero?

The Attempt at a Solution

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I know it's solution uses differentiation and i know it's probably easy, but i can't seem to get a grasp at it, and it really frustrates me. I've done it manually and is ≅50 weeks, but that didn't satisfied me and i need that answer nice and tidy using calculus. Please help me out. Thanks in advance

 

[Ray Vickson]

Homework Statement

[/B]
Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is:
P_n = P_n-1-√P_n-1

Homework Equations

[/B]
Considering that the civilization starts with a population of 676 people, how many week until it reaches zero?

The Attempt at a Solution

[/B]
I know it's solution uses differentiation and i know it's probably easy, but i can't seem to get a grasp at it, and it really frustrates me. I've done it manually and is ≅50 weeks, but that didn't satisfied me and i need that answer nice and tidy using calculus. Please help me out. Thanks in advance

Your model is a discrete-time difference model, so calculus is not really appropriate for it. Doing it manually is absolutely the correct way to go.

If you really do want to use calculus to get an approximate (not exact) answer, you can solve the differential equation
$$ \frac{dP}{dt} = -\sqrt{P}, \; P(0) = 676$$
and determine where $P(t) = 0.$

The real problem is that neither the discrete-time difference method of the continuous-time calculus method captures reality---the fact that at any time $P(n)$ or $P(t)$ must be an integer. Thus, the equation $P(n+1) = P(n) - \sqrt{P(n)}$ cannot hold exactly because it often produces a non-integer result for $P(n+1)$. It is not at all clear how to fix this up.

 

[Math Henry]

The real problem is that neither the discrete-time difference method of the continuous-time calculus method captures reality---the fact that at any time $P(n)$ or $P(t)$ must be an integer. Thus, the equation $P(n+1) = P(n) - \sqrt{P(n)}$ cannot hold exactly because it often produces a non-integer result for $P(n+1)$. It is not at all clear how to fix this up.

Thank you Ray for the quick reply, at least now i know that the answer is not that obvious.

If you really do want to use calculus to get an approximate (not exact) answer, you can solve the differential equation
$$ \frac{dP}{dt} = -\sqrt{P}, \; P(0) = 676$$
and determine where $P(t) = 0.$

So, if I ignore the fact that I am using a model of population which can only hold meaning for integer populations, that is, any result P ∈ ℕ, and instead treat it like an ordinary system which accept results P ∈ ℝ , i get, by integrating both sides:
$$ P(t) = 676 - \sqrt{P}*t $$
Is that the furthest i can go in this problem with calculus? P.S.: I know that i can solve that kind of problem easily with a simple Python or FORTRAN program for bigger populations, but the book it never mentioned the need for it.

 

[Ray Vickson]

Thank you Ray for the quick reply, at least now i know that the answer is not that obvious.
So, if I ignore the fact that I am using a model of population which can only hold meaning for integer populations, that is, any result P ∈ ℕ, and instead treat it like an ordinary system which accept results P ∈ ℝ , i get, by integrating both sides:
$$ P(t) = 676 - \sqrt{P}*t $$
Is that the furthest i can go in this problem with calculus?P.S.: I know that i can solve that kind of problem easily with a simple Python or FORTRAN program for bigger populations, but the book it never mentioned the need for it.

No, you do NOT get $P(t) = 676 - \sqrt{P(t)} t$; that is not how you solve a differential equation. Anyway, your notaton $\sqrt{P} * t$ is meaningless, since there is only one $P$ here---and it takes $t$ as an argument. When we write a differential equation as $dP/dt =-\sqrt{P}$ that is really shorthand notation for $dP(t)/dt = -\sqrt{P(t)}$.

Google "differential equation" for more information, or look in your textbook if you have one (and if it covers that topic). Even better: Google "separable differential equation".