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A tank with a capacity of 100 gallons initially contains 50 gallons of

A tank with a capacity of 100 gallons initially contains 50 gallons of water with 10 pounds of salt in solution. fresh water enters at a rate of 2 gallons per minute and a well-stirred mixture is pumped out at the rate of 1 gallon per minute. compute the amount of salt (in pounds) in the tank at the first moment when the tank is filled.

2 months ago

Solution 1

Guest Guest #9861
2 months ago
Let A(t) be the amount of salt (in pounds) in the tank at time t. We're given thatA(0)=10\text{ lb}.

The rate at which the amount of salt in the tank changes is given by the ODE

A'(t)=\dfrac{2\text{ gal}}{1\text{ min}}\cdot\dfrac{0\text{ lb}}{1\text{ gal}}-\dfrac{1\text{ gal}}{1\text{ min}}\cdot\dfrac{A(t)\text{ lb}}{50+(2-1)t\text{ gal}}

A'(t)+\dfrac{A(t)}{50+t}=0

(50+t)A'(t)+A(t)=0

\bigg((50+t)A(t)\bigg)'=0

(50+t)A(t)=C

A(t)=\dfrac C{50+t}

Given that A(0)=10, we find that

10=\dfrac C{50+0}\implies C=500

so that the amount of salt in the tank is described by

A(t)=\dfrac{500}{50+t}

The tank will be filled when 50+t=100, or after t=50 minutes. At this time, the amount of salt in the tank is

A(50)=\dfrac{500}{50+50}=5\text{ lb}

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