Generalize the given model in the equation:

$$ frac {{dA}} {{dt}} = 6 – frac {A} {{100}}

% MathType! MTEF! 2! 1! + –

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% daWcaaqaaiaadgeaaeaacaaIXaGaaGimaiaaicdaaaaaaaa! 3F4F!

$$ or

$$ frac {{dA}} {{dt}} + frac {1} {{100}} A = 6

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% aaqaaiaaigdacaaIWaGaaGimaaaacaWGbbGaeyypa0JaaGOnaaaa! 3FFF!

$$

The equation also goes on to explain that:

If rin and confuse the general flow of entry and exit brine solutions *, then

there are three possibilities: rin = rout, rin> rout and rin

< rout. In the analysis leading
to (8) we have assumed that rin=rout. In the latter two cases the number of gallons
of brine in the tank is either increasing (rin > rout) or decreasing (rin

<der) to

the net rate rin-rout.

assuming that the large reservoir initially contains N0

number of gallons of brine, Rin and Rout are the entrance and

brine outlet rate, respectively (measured in gallons

per minute), cin is the concentration of salt in

the influx, c (t) the concentration of salt in the reservoir

as well as in the outgoing flow at time t (measured in pounds)

of salt per gallon), and A (t) is the amount of salt in the

tank at time t.

or $$ c (t) = frac {{A (t)}} {{300}}

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% cacaWG0bGaaiykaiabg2da9maalaaabaGaamyqaiaacIcacaWG0bGa

% aiykaaqaaiaaiodacaaIWaGaaGimaaaaaaaa! 3F8F!

$$ (lb per gallon)

First, in an attempt to generalize, I concluded

(1) $$ Rin = (cin) (c (t))

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% OaGaai4yaiaacIcacaWG0bGaaiykaiaacMcaaaa! 4267!

$$

And

(2)

$$ Rout = frac {{A (t)}} {{c (t)}}

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% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x

% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaad +

% gacaWG1bGaamiDaiabg2da9maalaaabaGaamyqaiaacIcacaWG0bGa

% aiykaaqaaiaadogacaGGOaGaamiDaiaacMcaaaaaaa! 411C!

$$

and so

because

$$ frac {{dA}} {{dt}} = Rin – Rout

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% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x

% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca

% WGKbGaamyqaaqaaiaadsgacaWG0baaaiabg2da9iaadkfacaWGPbGa

% amOBaiabgkHiTiaadkfacaWGVbGaamyDaiaadshaaaa! 4200!

$$

is the general formula that I've used in every problem (where Rin = salt input rate, Rout = salt output rate) I thought that plugging this general data in case of a problem would make it a problem. but it turns out that the solution of the book is very different

(See picture)

I have trouble understanding how they brought their solution closer to what they gave us in the question and the equation. Any help to help me cross again would be a major help. Thanks in advance!