A pond initially contains 1,000,000 gal of water and an...

A pond initially contains $1,000,000$ gal of water and an unknown amount of an undesirable chemical. Water containing 0.01 gram of this chemical per gallon flows into the pond at a rate of $300 \mathrm{gal} / \mathrm{min}$. 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 whose solution is the amount of chemical in the pond at any time (b) How much of the chemical will be in the pond after a very long time? Does this limiting amount depend on the amount that was present initially?

A pond initially contains $1,000,000$ gal of water and an unknown amount of an undesirable
chemical. Water containing 0.01 gram of this chemical per gallon flows into the pond at a rate of $300 \mathrm{gal} / \mathrm{min}$. 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 whose solution is the amount of chemical in the pond at
any time
(b) How much of the chemical will be in the pond after a very long time? Does this limiting
amount depend on the amount that was present initially?

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They are used to model systems in which quantities change over time, capturing the dynamics of processes such as rates of growth or decay, mixing, and chemical reactions.

Mixing Problems

Mixing problems involve tracking the concentration or amount of a substance within a medium that is being mixed continuously. These problems typically use rates of inflow and outflow of both the substance and the medium to set up a model that predicts how the concentration evolves over time.

First-Order Linear Differential Equations

A first-order linear differential equation is an equation that contains the first derivative of an unknown function and can be written in the standard form. Such equations are prevalent in modeling mixing problems because they describe how the rate of change of a substance's amount depends linearly on its current amount and external inputs or removals.

Equilibrium or Steady-State Solution

The equilibrium or steady-state solution refers to the long-term behavior of a system described by a differential equation, where the rate of change becomes zero. In the context of mixing problems, it indicates the eventual concentration or amount of a substance in the mixture after the effects of initial conditions decay over time.

Uniform Distribution

Uniform distribution in the context of mixing means that the substance is evenly spread throughout the medium at all times. This assumption simplifies the analysis of mixing problems because it ensures that the concentration throughout the entire volume is the same, allowing for straightforward application of conservation principles.

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