Suppose a tank contains 5 gallons of water and 1 pound of salt. The tank begins draining at 2 gallons per minute as fresh water is added at 1 gallon per minute.

Question

Suppose a tank contains 5 gallons of water and 1 pound of salt. The tank begins draining at 2 gallons per minute as fresh water is added at 1 gallon per minute. How much salt is in the tank after 4 minutes?

Answer 1

Let y be the number of pounds of salt in the water

dy/dt = (salt rate in)-(salt rate out)

salt rate in = 0/g (1 g/min) = 0

salt rate out = (y lb)/((5-t) g) × (2 g/min) = (2/(5-t)) y g/min Because we are losing 1 gallon per minute.

dy/dt = -2y/(5-t)

dy/y = -2/(5-t)

∫dy/y = -∫2/(5-t)dt

ln|y| = 2ln(5-t)+C₁

y = C₂(5-t)²

At t=0 y = 1 lb, So C₂=1/25

y =(5-t)²/25

After 4 minutes

y(4) = 1²/25 = 0.04 pounds