1 point) Initially 10 grams of salt are dissolved into 25 liters of water. Brine

Question

1 point) Initially 10 grams of salt are dissolved into 25 liters of water. Brine with concentration of salt 5 grams per liter is added at a rate of 3 liters per minute. The tank is well mixed and drained at 3 liters per minute. a. Let z be the amount of salt, in grams, in the solution after t minutes have elapsed. Find a formula for the rate of change in the amount of salt, dr/dt, in terms of the amount of salt in the solution z dr grams/minute dt b. Find a formula for the amount of salt, in grams, after t minutes have elapsed. z(t) = grams c. How long must the process continue until there are exactly 20 grams of salt in the tank? minutes

Explanation / Answer

a)

x(0)=10

dx=5*3*dt-(x/25)*3dt

dx/dt=3(5-x/25)

b)

dx/(125-x)=3dt/25

Integrating gives

-ln(125-x)=3t/25+A

125-x=C exp(-3t/25)

x=125-C exp(-3t/25)

x(0)=125-C=10

C=115

x=125-115 exp(-3t/25)

c)

x(t)=20=125-115 exp(-3t/25)

Solving gives

t~0.758 minutes

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