A person is trying to fill her new swimming pool for the first time. Water is flowing into the pool at a constant rate of 30 liters/min. Unfortunately there is a hole and a crack in the pool from which water is leaking. Water is leaking out of the hole at a rate proportional to the square of the amount of water currently in the pool while water is leaking out of the crack at a rate proportional to the amount of water currently in the pool. As it had rained the previous day, the pool already contained 20 liters of water before filling began. let t(mins) be the time since the person started filling the pool and let W(t) be the number of liters of water in the pool at time t. Write down but do not solve the differential equation for W(t) along with its initial condition.

Question

A person is trying to fill her new swimming pool for the first time. Water is flowing into the pool at a constant rate of 30 liters/min. Unfortunately there is a hole and a crack in the pool from which water is leaking. Water is leaking out of the hole at a rate proportional to the square of the amount of water currently in the pool while water is leaking out of the crack at a rate proportional to the amount of water currently in the pool. As it had rained the previous day, the pool already contained 20 liters of water before filling began. let t(mins) be the time since the person started filling the pool and let W(t) be the number of liters of water in the pool at time t. Write down but do not solve the differential equation for W(t) along with its initial condition.

Now, is this correct?, this is as far as I have gotten but I am not sure if this is right?

I put dw/dt in as 30t, because it is 30 liters/min the rate of flow into the pool.

I put dw/dt out as -kw^2 - pw, because it is the sqaure and the constant amount as the rate of the flow out of the pool.

So combining this, dw/dt is 30t - kw^2 - pw where w(0) = 20 as the initial condition.

IS THIS RIGHT???

Thank you for the help in advance.

Answer 1

Yes, you're on the right track! Let's break down the problem and go over the correct formulation of the differential equation along with its initial condition.

Firstly, we have the rate at which water is flowing into the pool, which is a constant rate of 30 liters/min. So the term for the inflow rate will be 30.

Secondly, we have the outflow of water due to the hole and the crack. The amount of water leaking out of the hole is proportional to the square of the amount of water in the pool, which can be represented by -K * w^2, where K is a constant of proportionality. Similarly, the amount of water leaking out of the crack is proportional to the amount of water in the pool, and can be represented by -Pw, where P is another constant of proportionality.

To write the differential equation, we can use the fact that the rate of change of water in the pool (dW/dt) is equal to the inflow rate minus the outflow rate. Therefore, the equation becomes:

dW/dt = 30 - K * W^2 - P * W

Now, for the initial condition, you correctly mentioned that the pool already contained 20 liters of water before filling began. So at time t = 0, the amount of water in the pool, W(0), is equal to 20. This gives us the initial condition:

W(0) = 20

Therefore, the correct formulation of the differential equation with its initial condition is:

dW/dt = 30 - K * W^2 - P * W
W(0) = 20

Now, you can proceed with solving this differential equation using appropriate methods such as separation of variables, substitution, or numerical methods to determine the function W(t) that represents the amount of water in the pool as a function of time.