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kodGreya [7K]
2 years ago
10

a tank contains 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 2 grams of salt per liter is then p

umped into the tank at a rate of 4 L/min. The well-mixed solution is pumped out at a slower rate of 3 L/min. Write a differential equation that models the amount of salt in the tank at any time.
Mathematics
1 answer:
myrzilka [38]2 years ago
5 0

Salt flows in at a rate of

(2 g/L) * (4 L/min) = 8 g/min

and out at a rate of

(<em>B</em>/(200 + <em>t</em>) g/L) * (3 L/min) = 3<em>B</em>/(200 + <em>t</em>) g/min

where <em>B</em> is the amount of salt in the tank at time <em>t</em>.

Then the net rate at which <em>B</em> changes is governed by the ODE,

B'=8-\dfrac{3B}{200+t}

B'+\dfrac{3B}{200+t}=8

Multipy both sides by (200+t)^3:

(200+t)^3B'+3B(200+t)^2=8(200+t)^3

\left(B(200+t)^3\right)'=8(200+t)^3

Integrate both sides:

B(200+t)^3=2(200+t)^4+C

B=2(200+t)+C(200+t)^{-3}=\dfrac{2(200+t)^4+C}{(200+t)^3}

The tank starts with 30 g of salt, so <em>B</em>(0) = 30, which gives

30=2(200) + C(200)^{-3}\implies C=-2,960,000,000

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