Final answer:
The differential equation for the concentration of salt in the tank over time is dy/dt = 0.0625 - (y/2000)*5, with an initial condition y(0) = 50 kg.
Step-by-step explanation:
The student is tasked with determining the differential equation that models the concentration of salt in a tank over time. The tank starts with 50 kg of salt and 2000L of water, and a salt solution flows in at a rate of 5L/min with a concentration of 0.0125kg of salt per liter. The mixed solution drains from the tank at the same rate it enters.
Part 1: Differential Equation for dy/dt
The rate of change of the amount of salt in the tank, denoted as dy/dt, is the difference between the rate at which salt enters and the rate at which it leaves the tank. Therefore, dy/dt is given by the equation:
dy/dt = rate in (kg/min) - rate out (kg/min)
The rate in is the concentration of the entering solution multiplied by the flow rate:
rate in = 0.0125 kg/L * 5 L/min = 0.0625 kg/min
The rate out is the concentration of salt in the tank at any time t, divided by the total volume, multiplied by the flow rate:
rate out = (y/2000 kg/L) * 5 L/min
Part 2: Initial Condition y(0)
The initial condition for the amount of salt in the tank at time t=0 minutes is:
y(0) = 50 kg