Differential Story on Chlorination and Evaporation

[Fritzy]

Homework Statement

A pool has chlorine-free water. The chlorinator is turned on, dissolving chlorine in the pool at a rate of 30g/hr. But the chlorine has an escape rate proportional to the amount dissolved in water. This pool's escape rate is 13g/hr per 100grams dissolved.

a. Write a differential equation that expresses this information and solve it to express the number of grams of chlorine in a pool as a function of the number of hours the chlorinator has been running. "Be Clever."

b. How long will it take for the chlorine to build up to the desired 200g?

Homework Equations

I believe I'm supposed to use dG/dt but it gets fuzzy from there.
dG/dt = _______ + C
and then from there we solve it so it looks like G(t)=Ce^kt

The Attempt at a Solution

We are solving for G(t) correct?
I think it is asking for a [rate in - rate out] kind of differential equation.
This is high school calculus so I believe it should be basic differential/ integration solving if that helps anyone?

 

[HallsofIvy]

Homework Statement

A pool has chlorine-free water. The chlorinator is turned on, dissolving chlorine in the pool at a rate of 30g/hr. But the chlorine has an escape rate proportional to the amount dissolved in water. This pool's escape rate is 13g/hr per 100grams dissolved.

a. Write a differential equation that expresses this information and solve it to express the number of grams of chlorine in a pool as a function of the number of hours the chlorinator has been running. "Be Clever."

b. How long will it take for the chlorine to build up to the desired 200g?

Homework Equations

I believe I'm supposed to use dG/dt but it gets fuzzy from there.
dG/dt = _______ + C
and then from there we solve it so it looks like G(t)=Ce^kt

First, until you say what "G" and "t" mean, this doesn't make any sense. It would make sense for G to be amount of chlorine in the pool or the concentration of chlorine per liter. Since you don't give a volume for the pool, I am going to guess that G is the amount of chlorine, in grams, in the entire pool and t is the time in hours. The "dG/dt" has units of grams (of chlorine) per hour. You are told that the choline comes into the pool at 30 g/hr and goes out of the pool at "13g/hr per 100grams dissolved" and G(t) (I am guessing) is the number of grams dissolved, that is -13(G/100).

The Attempt at a Solution

We are solving for G(t) correct?

No, the question was "How long will it take for the chlorine to build up to the desired 200g?" so they are asking for a time. However, you might want to solve for G(t) to find that.

I think it is asking for a [rate in - rate out] kind of differential equation.
This is high school calculus so I believe it should be basic differential/ integration solving if that helps anyone?

Yes, using the "rate in" and "rate out" I give above. This is a relatively simple, linear, first order, differential equation which can be solved by a simple integration.

 

[Fritzy]

First, until you say what "G" and "t" mean, this doesn't make any sense. It would make sense for G to be amount of chlorine in the pool or the concentration of chlorine per liter. Since you don't give a volume for the pool, I am going to guess that G is the amount of chlorine, in grams, in the entire pool and t is the time in hours. The "dG/dt" has units of grams (of chlorine) per hour. You are told that the choline comes into the pool at 30 g/hr and goes out of the pool at "13g/hr per 100grams dissolved" and G(t) (I am guessing) is the number of grams dissolved, that is -13(G/100).


No, the question was "How long will it take for the chlorine to build up to the desired 200g?" so they are asking for a time. However, you might want to solve for G(t) to find that.


Yes, using the "rate in" and "rate out" I give above. This is a relatively simple, linear, first order, differential equation which can be solved by a simple integration.

I know this doesn't make sense. That is why I am asking ;)

and part A. is asking for the differention equation FIRST. Part b. is just my teacher asking me to solve it for a particular number. But she is primarily looking for the differential equation and my steps of integrating it. SO just ignore part b for now.

My problem is that I am not sure how to use the things/ variables I have been given to solve for it. That is why I need help. Sorry if it wasn't clear before.

 

[Fritzy]

And I'm sorry, t would be time and G would be grams of chlorine.