A tank initially contains 900 liters of pure water. Then water containing 6 mg of detergent per liter starts to enter the tank at the rate of 20 liters per hour. (a) How long until the average concentration of detergent in the tank is 2 mg per liter?

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05-11-2019
Mathematics

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A tank initially contains 900 liters of pure water. Then water containing 6 mg of detergent per liter starts to enter the tank at the rate of 20 liters per hour. (a) How long until the average concentration of detergent in the tank is 2 mg per liter?

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Ava45018 Ava45018 · Answer 1 · 2019-11-05T20:52:08+01:00

A tank initially contains 1000 liters of pure water. Then water containing 4 mg of detergent per liter starts to enter the tank at the rate of 30 liters per hour.

Let; / ( if; )

Volume at any time t = V(t)

Concentration at any time t = C(t)

Amount at any time = Q(t)

(then;)

C(t) = Q(t) / V(t)

dV/dt = 30

V(t) = 30t + 1000

dQ/dt = (4 )(30)

dQ = 120 dt

Q = 120t

(a) How long until the average concentration of detergent in the tank is 2 mg per liter?

____hours

C(t) = Q(t) / V(t)

C(t) = (120t) / ( 30t + 1000)

when C(t) = 2 mg/hr

(120t) / ( 30t + 1000) = 2

60t + 2000 = 120t

t = 100/3 hrs

t = 33 ⅓ hours

——————- ( 1 day 9 hours 20 minutes )

(b) How long until the average concentration of detergent in the tank is x mg per liter?

(120t) / ( 30t + 1000) = x

30xt + 1000x = 120t

3xt + 100x = 12t

12t - 3xt = 100x

t = 100x/(12 - 3x)