A 300-liter tank initially holds a solution containing 50 kg of dissolved dye
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A 300-liter tank initially holds a solution containing 50 kg of dissolved dye. Beginning at t=0 minutes, two inlet pipes are opened:
• Pipe A brings in dye solution at a concentration of 0.80 kg L−1 and a flow rate of 4 L min−1.
• Pipe B brings in pure water at a flow rate of 2 L min−1.
At the same time, the mixture is drained from the tank at a rate of 6 L min−1. The tank volume therefore remains constant at 300 liters, and perfect mixing is assumed.
(a) Derive a differential equation for Q(t), the mass of dye (in kg) in the tank at time t.
(b) Solve the differential equation with the initial condition Q(0)=50 to obtain an explicit formula for Q(t).
(c) Compute the concentration of dye, in kg L−1, after one hour of operation. Give your answer to three significant figures.
(d) The manufacturer must reduce the dye concentration to below 0.10 kg L−1 before discharging the liquid. How long (in minutes) must the system operate to meet this requirement?