Question

Differential equations. please show work and explain each step! Youre are in charge of a paper mill. Your company got a contract to make a few rolls of paper of a particular kind. You need to add a chemical to a tank already containing that chemical in order to enhance the whiteness of the paper.

Consider you cannot stop the mill to add the chemical. The tank receives some input at 20 L per minutes and feed the mill at the same rate. If the tank as a volume of 3000 L, that initially there is 0.3 kg of chemical per L in the tank, and that the added compound must be at a fixed concentration, what should be that concentration if you want the output of the tank to become at 0.7 kg per L in exactly 120 minutes of operation?

Answer 1

To enhance the whiteness of the paper, we can use differential equations to determine the required concentration of the added chemical in the tank. By setting up a differential equation based on the input and output rates, we can solve for the concentration at a specific time. Applying the given initial condition and integrating, we can find the concentration in the tank after 120 minutes of operation.

Step-by-step explanation:

To solve this problem, we can use the concept of differential equations. Let's denote the concentration of the chemical in the tank as y(t), where t is the time in minutes.

The tank receives an input of 20 L per minute, so the rate of change of the concentration in the tank is equal to the input rate minus the output rate. This can be expressed as:

dy/dt = (20 - y(t))/3000

Now, we need to solve this differential equation to find the concentration y(t) at a specific time. We can separate variables and integrate:

∫ (3000/(20 - y(t))) dy = ∫ dt

By integrating, we get:

3000 ln|20 - y(t)| = t + C

Now, we can apply the initial condition that at t=0, y(0) = 0.3 kg/L. Substituting these values, we can solve for C:

3000 ln|20 - 0.3| = 0 + C

Now, we can solve for C:

3000 ln|19.7| = C

Finally, we can solve for y(t) when t=120 minutes:

3000 ln|20 - y(120)| = 120 + 3000 ln|19.7|

Now, solve for y(120) to find the concentration in the tank after 120 minutes of operation.