Salt problem

**lord12** · January 28th 2009, 12:06 PM

A tank with capacity of 400 gal of water originally contains 200 gal of water with 100 lb of salt in solution. Water containing 1 lb of salt per gallon is entering at a rate of 3 gal/ min, and the mixture is allowed to flow out of the tank at a rate of 2 gal/ min. Let Q(t) lb. be the amount of salt in the tank, V(t) gal be the volume of water in the tank.
Find the amount of salt in the tank at any time prior to the instant when the solution begins to overflow. Your answer should be a function in terms of t.

Doesn't Q(T) = t +200?

Find the concentration (in pounds per gallon) of salt in the tank when it is on the point of overflowing. Round your answer to 3 decimal places.

Doesn't this just equal 200 +200 = 400?

**Danny** · January 28th 2009, 12:44 PM

Originally Posted by lord12

A tank with capacity of 400 gal of water originally contains 200 gal of water with 100 lb of salt in solution. Water containing 1 lb of salt per gallon is entering at a rate of 3 gal/ min, and the mixture is allowed to flow out of the tank at a rate of 2 gal/ min. Let Q(t) lb. be the amount of salt in the tank, V(t) gal be the volume of water in the tank.
Find the amount of salt in the tank at any time prior to the instant when the solution begins to overflow. Your answer should be a function in terms of t.

Doesn't Q(T) = t +200?

Find the concentration (in pounds per gallon) of salt in the tank when it is on the point of overflowing. Round your answer to 3 decimal places.

Doesn't this just equal 200 +200 = 400?

Well if you think about it, initially you have 100 lbs of salt in 200 gallons or 1/2 lb/gal. The concentration of the incoming salt solution is 1 lb/gal. So if you assume perfect (and instanteous) mixing, the concentration will change from 1/2 to 1 (assuming an infinite tank). Here we must set up a differential equation for the process.

First the tank in fill up $V =(3-2)t + 200$ .

Next, the change in the amount of salt A is

$\frac{dA}{dt} = r_i c_i - r_o c_o$

where ri - rate in, ro - rate out, ci - concentration in and co - concentration out.

so

$\frac{dA}{dt} = 3 \cdot 1 - 2 \cdot \frac{A}{V} = 3 - \frac{2A}{t+200}$

the initial condition

$A(0) = 100$ . A linear ODE for A. Can you solve it?

**lord12** · January 28th 2009, 07:37 PM

I solve the diffeq and get that A = t +200

**Danny** · January 28th 2009, 07:46 PM

Originally Posted by lord12

I solve the diffeq and get that A = t +200

Please, let me ask - how?

**lord12** · January 28th 2009, 08:57 PM

is it t-100?

**Danny** · January 29th 2009, 05:00 AM

Originally Posted by lord12

is it t-100?

I got

$A = t + 200 - \frac{4,000,000}{(t+200)^2}$ .

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