The concentration of pollutant in a lake is given by the...

The concentration of pollutant in a lake is given by the equation $$ \frac{d c}{d t}=-0.055 c, $$ where $c$ is the concentration of the pollutant at $t$ days. Suppose that the initial concentration of pollutant is $0.10$. A concentration level of $c=0.02$ is deemed safe for the fish population in the lake. If the concentration varies according to the model, how long will it be before the concentration reaches a level that is safe for the fish population?

The concentration of pollutant in a lake is given by the equation
$$
\frac{d c}{d t}=-0.055 c,
$$
where $c$ is the concentration of the pollutant at $t$ days. Suppose that the initial concentration of pollutant is $0.10$. A concentration level of $c=0.02$ is deemed safe for the fish population in the lake. If the concentration varies according to the model, how long will it be before the concentration reaches a level that is safe for the fish population?

Key Concepts

Logarithmic Functions in Exponential Equations

Logarithms are used to solve for variables located in the exponent when working with exponential equations. They provide a way to isolate time or other parameters by converting multiplicative relationships into additive ones, which is especially important when determining the exact moment when a system reaches a specific condition.

Separation of Variables

Separation of variables is a common technique for solving first-order differential equations. It involves rearranging the equation so that each variable and its differential appear on opposite sides of the equation, allowing for straightforward integration. This method is particularly useful in problems involving exponential growth or decay.

Initial Value Problems

Initial value problems are differential equations accompanied by specific conditions that define the state of the system at a particular point in time. These conditions are essential for determining the unique solution to the problem by fixing the constants that arise from the integration process.

Ordinary Differential Equations

Ordinary differential equations (ODEs) are mathematical equations that involve functions of a single independent variable and their derivatives. They are widely used to model dynamic processes where the rate of change of a quantity is related to the quantity itself, making them essential for understanding and predicting various natural phenomena.

Exponential Decay

Exponential decay is a process where the rate of decrease of a quantity is proportional to its current value. This concept is fundamental in many areas such as physics, chemistry, and finance, as it describes how substances or values decrease rapidly at first and then more slowly over time under constant proportional loss.

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