Problem 8 - Applied Calculus

Question

The concentration of carbon dioxide, \(C=f(t),\) in the atmosphere, in parts per million (ppm), is a function of years, \(t,\) since 1960 (a) Interpret \(f(40)=370\) in terms of carbon dioxide." (b) What is the meaning of \(f(50) ?\)

Step-by-Step Solution

Answer

(a) In 2000, CO2 concentration was 370 ppm. (b) \(f(50)\) is CO2 concentration in 2010.

Step 1: Understanding the Given Function

The exercise uses the function \(C = f(t)\) to represent the concentration of carbon dioxide in parts per million, with \(t\) representing years since 1960. This means that \(t = 0\) corresponds to the year 1960.

Step 2: Interpreting f(40)=370

The notation \(f(40) = 370\) means that when \(t = 40\), which corresponds to the year 1960 + 40 = 2000, the concentration of carbon dioxide was 370 ppm. In terms of carbon dioxide, in the year 2000, the concentration of carbon dioxide in the atmosphere was 370 ppm.

Step 3: Understanding the Meaning of f(50)

\(f(50)\) represents the concentration of carbon dioxide in parts per million after 50 years from 1960, which corresponds to the year 1960 + 50 = 2010. Thus, \(f(50)\) gives us the carbon dioxide concentration in the atmosphere for the year 2010.

Key Concepts

Function Interpretation

Understanding how to interpret functions is crucial in mathematics, particularly when dealing with real-world applications like atmospheric science. In this exercise, we are given a function, \(C = f(t)\), which describes how the concentration of carbon dioxide changes over time.
This function is defined such that \(t\) represents the number of years since 1960, making it easy to determine the year that corresponds to different values of \(t\).

• \(f(40) = 370\) tells us that in the year 2000 (1960 + 40), the carbon dioxide concentration was 370 ppm.
• Similarly, \(f(50)\) refers to the concentration in 2010 (1960 + 50).

Being able to interpret these functions helps us make sense of data points in a broader context.

Parts Per Million

Parts per million (ppm) is a unit of measurement used to express very diluted concentrations of substances. It is particularly useful in environmental science to describe concentrations of gases in the atmosphere like carbon dioxide.
One ppm means that there is one part of a substance for every one million parts of the medium it is in, which in this case is air.

• If \(f(t) = 370\) ppm, it signifies that out of every million parts of atmospheric air, 370 are carbon dioxide molecules.
• This unit allows scientists to easily track small but significant changes in atmospheric composition over time.

Understanding ppm is essential when interpreting and discussing atmospheric data.

Atmospheric Science

Atmospheric science focuses on the study of the Earth's atmosphere and its various components, including gases like carbon dioxide. Understanding the concentration of greenhouse gases is critical for analyzing climate change and environmental health.
By analyzing trends in carbon dioxide levels using functions like \(f(t)\), scientists can make predictions about future climate conditions.

• The study of these trends allows for informed decision-making regarding environmental policy and protection measures.
• Real-world data is crucial in building models that help us understand the long-term effects of increasing carbon dioxide levels.

Atmospheric science combines data analysis with scientific theory to address present and future environmental challenges.

Calculus Applications

Calculus provides tools to model and analyze changes in natural systems, making it invaluable in understanding atmospheric data. For instance, by using derivatives and integrals, scientists can discern rates of change in carbon dioxide concentrations over time.
This exercise, although basic, lays the groundwork for more complex analysis, such as:

• Determining the growth rate of carbon dioxide concentrations by differentiating \(f(t)\).
• Calculating the total increase in carbon dioxide over a period by integrating \(f(t)\).

These procedures allow for a deeper understanding of how atmospheric conditions evolve, aiding in the formulation of responses to environmental impacts.