# Solving a Chemical Distribution Problem in a Pond Using Differential Equations

• ehabmozart
In summary, the differential equation for the amount of chemical in the pond at any time is given by dq/dt = 300 - 300q(t)/10^6, where q(t) is the amount of chemical in the pond at time t and 300 is the rate at which water containing 0.01 g of the chemical per gallon flows into the pond.
ehabmozart
1. Homework Statement

Apond initially contains 1,000,000 gal of water and an unknown amount of an undesirable
chemical. Water containing 0.01 g of this chemical per gallon flows into the pond at a rate
of 300 gal/h. The mixture flows out at the same rate, so the amount of water in the pond
remains constant. Assume that the chemical is uniformly distributed throughout the pond.

(a) Write a differential equation for the amount of chemical in the pond at any time.

## Homework Equations

This should be done without a prerequisite of any formula

## The Attempt at a Solution

Well, I do have the solution which says let q(t) be the amount chemical at any time. Hence, the concentration of this chemical in the water at any time is q(t) / 1000000. (This part is understood). 300*(0.01) is the amount of chemical coming into the pond every hour. (This part is fine too).. Now, the last and most important part ... 300 q(t) / 100000 is the rate at which the chemical leaves the pond per hour. What?? How did they give this claim?? I mean, in the text it is written that " The mixture flows out at the same rate, so the amount of water in the pond
remains constant. ". I am completely perplexed here. Honestly, i need to know some basics of DE. If anyone would give me a hand on how to approach the answer, I would owe him/her big time. Thanks a lot for your time!

ehabmozart said:
1. Homework Statement

Apond initially contains 1,000,000 gal of water and an unknown amount of an undesirable
chemical. Water containing 0.01 g of this chemical per gallon flows into the pond at a rate
of 300 gal/h. The mixture flows out at the same rate, so the amount of water in the pond
remains constant. Assume that the chemical is uniformly distributed throughout the pond.

(a) Write a differential equation for the amount of chemical in the pond at any time.

## Homework Equations

This should be done without a prerequisite of any formula

## The Attempt at a Solution

Well, I do have the solution which says let q(t) be the amount chemical at any time. Hence, the concentration of this chemical in the water at any time is q(t) / 1000000. (This part is understood). 300*(0.01) is the amount of chemical coming into the pond every hour. (This part is fine too).. Now, the last and most important part ... 300 q(t) / 100000 is the rate at which the chemical leaves the pond per hour. What?? How did they give this claim?? I mean, in the text it is written that " The mixture flows out at the same rate, so the amount of water in the pond
remains constant. ". I am completely perplexed here. Honestly, i need to know some basics of DE. If anyone would give me a hand on how to approach the answer, I would owe him/her big time. Thanks a lot for your time!

"The mixture flows out at the same rate, so the amount of water in the pond remains constant" means that the rate at which water flows out is the same as the rate at which it flows in, ie. 300 gallons per hour. But it is nowhere said that the chemical leaves at the same concentration as it enters. Instead the assumption is that the chemical is uniformly distributed throughout the pond, so that concentration at the outflow is the same as it is everywhere in the pond, ie. $q(t)/10^6$. The rate at which it leaves is then $300q(t)/10^6$.

## 1. What is a "weird question" in differential equations?

A weird question in differential equations is one that may seem unusual or unconventional, but is still relevant to the study and application of differential equations. It could also refer to a question that has a strange or unexpected solution.

## 2. How do you approach solving a weird question in differential equations?

The approach to solving a weird question in differential equations is similar to solving any other type of question. It involves understanding the problem, identifying the variables and equations involved, and applying appropriate mathematical techniques to find a solution. However, since weird questions may not have straightforward solutions, it may require creative thinking and alternative approaches.

## 3. Can a weird question in differential equations have practical applications?

Yes, a weird question in differential equations can have practical applications. In fact, many real-world problems can be modeled using differential equations and may result in unusual or unexpected solutions. These solutions can provide valuable insights and help in understanding complex systems.

## 4. Are there any specific techniques or methods for solving weird questions in differential equations?

There is no specific set of techniques or methods for solving weird questions in differential equations. As with any problem, different approaches may be required depending on the nature of the question. However, having a strong understanding of differential equations and mathematical principles can help in approaching and solving these types of questions.

## 5. What can studying weird questions in differential equations reveal about the field of mathematics?

Studying weird questions in differential equations can reveal the versatility and complexity of the field of mathematics. It can also demonstrate the importance of creative thinking and problem-solving skills in tackling unconventional problems. Additionally, it can lead to new discoveries and advancements in the study of differential equations and their applications.

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