Solving Exponential Growth Homework: Find Water to Eliminate 50% Salt

[lovemake1]

Homework Statement

Water is pumped into a tank. Volume V, is kept constant by continuos flow. The amount of salt S, depends on the amount of water that ahs been pumped in, call it X.

ds/dx = -S/V

Find the amount of water needed to eliminate 50% of the salt. Take v AS 10,000 gallons

Homework Equations

The Attempt at a Solution

We know that volume V is kept constant by continuos flow. d/dv ?
im througly confused...
I understand that exponential growth has the traditional formula: Ce^kt
but how can we ues it with this question?

helps appreciated.

 

[HallsofIvy]

Homework Statement

Water is pumped into a tank. Volume V, is kept constant by continuos flow. The amount of salt S, depends on the amount of water that ahs been pumped in, call it X.

ds/dx = -S/V

Find the amount of water needed to eliminate 50% of the salt. Take v AS 10,000 gallons

Homework Equations

The Attempt at a Solution

We know that volume V is kept constant by continuos flow. d/dv ?
im througly confused...

So am I! There is no differentiation with respect to V (please do not use both small and capital letters to mean the same thing). V is a constant- replace it with the 10000 you are given.

I understand that exponential growth has the traditional formula: Ce^kt
but how can we ues it with this question?

helps appreciated.

I take it this is a "differential equations" course. You would be expected to recognize that you can write the original equation as
[tex]\frac{dS}{S}= -\frac{dx}{10000}[/tex]
and integrate:
[tex]ln(S)= -\frac{x}{10000}+ C[/tex]
so that
[tex]S(x)= e^C e^{-\frac{x}{10000}}= C' e^{-\frac{x}{V}[/itex]
where [itex]C'= e^C[/itex]. That is the "[itex]Ce^{kt}[/itex]" you have, except that your variable is x, the amount of water that has flowed through, not time.

Now solve the equation S(x)= (1/2)S(0).

 

[lovemake1]

Ah I see the light !
But the line after "so that",
Shouldn't it be e^lns = e^-x/v + c ?
just confused to what C is