Consider a 50 gallon tank containing 20 gallons of water with 10 pounds of salt

Question

Consider a 50 gallon tank containing 20 gallons of water with 10 pounds of salt mixed in. Beginning at time t = 0 brine (salt water) with a salt concentration of 1 pound per gallon flows into the tank at a constant rate of 3 gallons per minute and the well-stirred mixture flows out of the tank at the 2 gallons per minute. Let S(t) denote the mass (Ibs) of salt in the tank at time t. Write down an initial-value problem that governs S(t) (Do not solve). Is the equation in part (a) Linear? Separable? Autonomous? List all that apply.

Explanation / Answer

a)rate of change of amount of salt in tank= rate of inflow of salt -rate of out flow of salt

rate of inflow of salt =1*3=3

rate of outflow of salt= S(t)/(20+(3-2)*t)

rate of change of amount of salt in tank= S'(t)=1*3 - S(t)/(20+(3-2)*t)

S'(t)=3 - S(t)/(20+t)

S(0)=10

b) equation is linear

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