Question

Suppose that a large mixing tank initially holds 400 gallons of water in which 70 pounds of salt have been dissolved. Pure water is pumped into the tank at a rate of 4 gal/min, and when the solution is well stirred, it is then pumped out at the same rate. Determine a differential equation for the amount of salt A(t) in the tank at time t > 0. What is A(0)

Answer 1

To determine the differential equation for the amount of salt in the tank at time t > 0, we consider the rate at which salt enters and leaves the tank. The differential equation is dA/dt = (0 - A(t))*(4/400). To find A(0), we substitute t = 0 into the equation.

Step-by-step explanation:

To determine the differential equation for the amount of salt A(t) in the tank at time t > 0, we need to consider the rate at which salt enters and leaves the tank. The rate at which salt enters the tank can be calculated by multiplying the concentration of salt in the incoming water (which is 0 pounds/gallon) by the rate at which water is pumped into the tank (which is 4 gallons/minute). The rate at which salt leaves the tank can be calculated by multiplying the concentration of salt in the tank (which is A(t) pounds/gallon) by the rate at which water is pumped out of the tank (which is also 4 gallons/minute). Therefore, the differential equation for the amount of salt A(t) in the tank is dA/dt = (0 - A(t))*(4/400).

To find A(0), we substitute t = 0 into the differential equation. The equation becomes dA/dt = (0 - A(0))*(4/400), which simplifies to dA/dt = -A(0)/100. To solve this separable differential equation, we can separate the variables and integrate both sides. Integrating gives us ln|A| = -t/100 + C, where C is the constant of integration. Exponentiating both sides gives us

, which simplifies to

, where K is a constant determined by the initial condition A = A(0).

Answer 2

`(dA)/(dt)+ (A)/(100)=0`

A(0) = 70 (i.e the pounds of salt dissolved into the tank)

Step-by-step explanation:

Given that:

a large mixing tank initially holds 400 gallons of water in which 70 pounds of salt have been dissolved.

The rate at which pure water is pumped into the tank is 4 gal/min

After stirring; the pure water is then pumped out at the same rate.

The objective is to determine the differential equation for the amount of salt A(t) in the tank at time t > 0. What is A(0)

Taking the differential of:

`(dA)/(dt)=R_(in)-R_(out)` ---- (1)

where ;

`R_{in` = 0

`R_(out) =(A(t))/(400)*4`

`R_(out) =(A)/(100)`

replacing them into (1) ; we have:

`(dA)/(dt)=0 - (A)/(100)`

`(dA)/(dt)=- (A)/(100)`

`(dA)/(dt)+ (A)/(100)=0`

A(0) = 70 (i.e the pounds of salt dissolved into the tank)