solve and show all steps to all parts A tank contains 100 gallons of water. 5 ga

Question

solve and show all steps to all parts

A tank contains 100 gallons of water. 5 gallons of brine per minute flow into the tank, each gallon of brine containing 2 pounds of salt. 5 gallons of water flow out of the tank per minute. assume that the tank is kept well served.

A) find a differential equation for the number of lbs of salt in the tank (call it A)

B) assume the tank initially contains 10 lbs of salt, solve this differential equation.

C) how much salt in the tank after 10 minutes?

D) at what time will there be 199 lbs of salt in the tank? give an exact numerical answer with 3 decimal places.

Explanation / Answer

dA/dt = 5*2 - 5*(A/100) = 10 - A/20

20 ln(10 - A/20) = -5t + ln c

at t=0, A = 10

20ln 9.5 = ln c

c = 9.5^20

ln(10 - A/20) = -t/4 + ln 9.5

10 - A/20 = 9.5e^{-t/4}

A = 20(10-9.5e^{-t/4})

at t = 10

A = 184.4 lbs

199 = 20(10 - 9.5e^{-t/4})

t = 20.988 min

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