Problem 42 - Algebra A Combined Function

Question

Solve. Unless otherwise indicated, round results to one decimal place. Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) is a greenhouse gas that contributes to global warming. Partially due to the combustion of fossil fuels, the amount of \(\mathrm{CO}_{2}\) in Earth's atmosphere has been increasing by \(0.4 \%\) annually over the past century. In \(2000,\) the concentration of \(\mathrm{CO}_{2}\) in the atmosphere was 369.4 parts per million by volume. To make the following predictions, use \(y=369.4(1.004)^{t}\) where \(y\) is the concentration of \(\mathrm{CO}_{2}\) in parts per million and \(t\) is the number of years after 2000. (Sources: Based on data from the United Nations Environment Programme and the Carbon Dioxide Information Analysis Center) a. Predict the concentration of \(\mathrm{CO}_{2}\) in the atmosphere in the year 2012 . b. Predict the concentration of \(\mathrm{CO}_{2}\) in the atmosphere in the year 2030 .

Step-by-Step Solution

Answer

The CO2 concentrations are approximately 387.7 ppm in 2012 and 416.3 ppm in 2030.

Step 1: Identify the formula variables

We will use the given formula \( y = 369.4(1.004)^t \) where \( y \) is the concentration of CO2 in parts per million and \( t \) is the number of years after 2000.

Step 2: Calculate years for predictions

For part (a), the year 2012 corresponds to \( t = 2012 - 2000 = 12 \). For part (b), the year 2030 corresponds to \( t = 2030 - 2000 = 30 \).

Step 3: Predict CO2 concentration for 2012

Substitute \( t = 12 \) into the formula: \[ y = 369.4(1.004)^{12} \] Calculate the result by evaluating \( 1.004^{12} \) and multiplying by 369.4.

Step 4: Calculate power and product for 2012

First, calculate \( 1.004^{12} \). This brings us to approximately 1.049. Then, multiply this by 369.4: \[ y = 369.4 imes 1.049 \approx 387.7 \] ppm.

Step 5: Predict CO2 concentration for 2030

Substitute \( t = 30 \) into the formula: \[ y = 369.4(1.004)^{30} \] Calculate the result by evaluating \( 1.004^{30} \) and multiplying by 369.4.

Step 6: Calculate power and product for 2030

First, calculate \( 1.004^{30} \). This brings us to approximately 1.127. Then, multiply this by 369.4: \[ y = 369.4 imes 1.127 \approx 416.3 \] ppm.

Key Concepts

Carbon Dioxide Concentration

Carbon dioxide concentration refers to the amount of carbon dioxide (\( \mathrm{CO}_2 \)) present in a certain volume of the Earth's atmosphere. CO2 is measured in parts per million by volume (ppm). For example, in the year 2000, the concentration of CO2 in the atmosphere was 369.4 ppm.
The concentration of carbon dioxide in the atmosphere is an essential factor in studying climate change. This number increases due to various human activities, especially the burning of fossil fuels for energy. Every year, there is an increase of approximately 0.4% in CO2 levels, compounding over time.
When modeling the growth of CO2 concentration over the years, assumptions like this percentage increase help in making predictions about future levels. These predictions allow scientists and policymakers to estimate the environmental impact of increasing CO2 concentrations and work towards potential solutions.

Greenhouse Gas Emissions

Greenhouse gases (GHGs), like carbon dioxide, methane, and nitrous oxide, trap heat in the Earth's atmosphere. This creates a "greenhouse effect," which leads to global warming. By far, carbon dioxide is the most significant contributor to man-made global warming.
Most CO2 emissions are the result of burning fossil fuels such as coal, oil, and natural gas. These fuels are often used in electricity generation, transportation, and various industrial processes. Additionally, other activities like deforestation and certain agricultural practices also increase CO2 levels.
The continuous emission of greenhouse gases imposes a severe threat to ecological balance. Rising temperatures lead to adverse consequences such as melting polar ice, rising sea levels, and increased frequency of extreme weather events. Thus, reducing greenhouse gas emissions is crucial for stabilizing the climate.

Mathematical Modeling

Mathematical modeling is a method used to represent real-world situations using mathematical formulas and concepts. The importance of modeling lies in its ability to predict future trends based on current data.
In the context of CO2 concentrations, mathematical models help us predict future amounts by considering current growth rates. For instance, the model given in the exercise equation \(y = 369.4(1.004)^t\) allows us to estimate CO2 levels at any future date.
This formula is an example of exponential growth, where something grows by a constant percentage each time period. The variable \( t \) represents the number of years since the baseline year 2000. Thus, by plugging in different values of \( t \), we can predict CO2 concentrations in different years. Such mathematical insights are invaluable for planning the timing and nature of environmental interventions.

Environmental Mathematics

Environmental mathematics utilizes mathematical methods and computational tools to address environmental issues. It is a branch of applied mathematics that involves modeling and analyzing ecological phenomena.
This field provides the framework to understand complex environmental interdependencies and predict outcomes. For example, modeling the increase in CO2 concentration is crucial for assessing its impact on climate.
By accurately predicting changes in environmental parameters, environmental mathematics assists in forming policies and conservation strategies. Beyond CO2 concentration, it encompasses other factors such as resource management, pollution levels, and ecosystem balance. Understanding these interactions is critical to pave the way for sustainable development.