Answer:
Explanation:
To derive the differential equation that describes the amount of salt in the tank, we need to use the law of conservation of mass. The rate of change of the amount of salt in the tank is given by the difference between the rate of salt entering the tank and the rate of salt leaving the tank. The rate of salt entering the tank is the product of the concentration of the solution and the flow rate of the solution, while the rate of salt leaving the tank is the product of the concentration of salt in the tank and the flow rate of the draining solution. Therefore, we get:
dy/dt = (0.6 kg/L x 7 L/min) - (y(t)/100 L x 5 L/min)
where y(t) is the amount of salt (in kilograms) in the tank after t minutes.
To find the amount of salt in the tank after 30 minutes, we can solve the differential equation we derived in part 1 using separation of variables:
dy/dt = 4.2 - 0.05y
dy/y - 4.2/0.05 dt = 0
Integrating both sides, we get:
ln|y| - 84t + C = 0
where C is the constant of integration. Solving for C using the initial condition that y(0) = 0, we get:
C = 84 ln(6)
Substituting back, we get:
ln|y| - 84t + 84 ln(6) = 0
ln|y| = 84t - 84 ln(6)
|y| = e^(84t - 84 ln(6))
Since y is the amount of salt, it must be positive, so we take the absolute value off and get:
y = e^(84t - 84 ln(6))
Plugging in t = 30, we get:
y = e^(84(30) - 84 ln(6)) ≈ 37.86 kg
Therefore, there is approximately 37.86 kg of salt in the tank after 30 minutes.