A large tank is filled to capacity with 300 gallons of pure water. Brine containing 5 pounds of salt per gallon is pumped into the tank at a rate of 3 gal/min. The well-mixed solution is pumped out at a rate of 6 gals/min. Find the number A(t) of pounds of salt in the tank at time t. A(t) = 1500(1−e−0.02t) lb How long (in minutes) will it take for the tank to be empty after this process has started? min

(a)

(b)

Step-by-step explanation:

Given

--- Volume of tank

--- Brine solution

 --- Rate in, in gallon/min

--- Rate out, in gallon/min

Required

Determine A(t)

First, calculate the rate in (R in) and (R out) in lb/min

R(out) is calculated by multiply the rate at which brine leaves the tank (lb/gal) * R2

So, we have:

The change in the amount of brin e in the tank at time t is given as:

Rewrite:

Multiply through by dt

Make dt stand alone

Express as exponents

Multiply through by 100

Apply initial conditions

Substitute 0 for t in:

Substitute 15 for c in

Factorize:

How long to empty the tank

Set A to 0

Divide through by 1500

Collect Like Terms

Take ln of both sides

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step-by-step explanation:

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