Answer to Question #273945 in Differential Equations for Peter

Question

A 50 gallons tank initially contains 10 gal of fresh water. At t = 0, a brine solution

containing 1 lb of salt per gallon is poured into the tank at the rate of 4 gal/min, while the

well-stirred mixture leaves the tank at the rate of 2 gal/min. Find

a. the amount of time required for overflow to occur

b. the amount of salt in the tank at the moment of overflow

Answer 1

2.

Let $s(t) =$ amount, in lb of salt at time $t.$ Then we have

$\dfrac{ds}{dt}=$(rate of salt into tank) − (rate of salt out of tank)

$$\dfrac{ds}{dt}=1(4)-\dfrac{2s}{10+(4-2)t}$$

So we get the differential equation

$$\dfrac{ds}{dt}+\dfrac{s}{5+t}=4, s(0)=0$$

Integrating factor

$$\mu=5+t$$

$$(5+t)\dfrac{ds}{dt}+s=4(5+t)$$

$$d((5+t)s)=4(5+t)dt$$

Integrate

$$\int d((5+t)s)=\int4(5+t)dt$$

$$(5+t)s=20t+2t^2+C$$

$$s=\dfrac{20t+2t^2+C}{5+t}$$

$$s(0)=\dfrac{C}{5}=0=>C=0$$

$$s(t)=\dfrac{20t+2t^2}{5+t}$$

a. $V=10+2t=50=>t=20$

The tank overflows at $t=20$ min.

b.

$$s(20)=\dfrac{20(20)+2(20)^2}{5+20}=48(lb)$$