Problem 43 - Calculus

Question

Concern biodiesel, a fuel derived from renewable resources such as food crops, algae, and animal oils. The table shows the percent growth over the previous year in US biodiesel consumption. \(^{14}\) $$\begin{array}{c|c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 \\ \hline \% \text { growth } & -12.5 & 92.9 & 237 & 186.6 & 37.2 & -11.7 & 7.3 \\\ \hline \end{array}$$ (a) According to the US Department of Energy, the US consumed 91 million gallons of biodiesel in 2005 Approximately how much biodiesel (in millions of gallons) did the US consume in \(2006 ?\) In \(2007 ?\) (b) Graph the points showing the annual US consumption of biodiesel, in millions of gallons of biodiesel, for the years 2005 to \(2009 .\) Label the scales on the horizontal and vertical axes.

Step-by-Step Solution

Answer

2006: 260.7 million gallons; 2007: 357.2 million gallons. Plot: 2005-91, 2006-260.7, 2007-357.2, 2008-315.2, 2009-338.6.

Step 1: Calculate 2006 Consumption

In 2005, the US consumed 91 million gallons of biodiesel. The growth rate in 2006 was 186.6%. To find the consumption for 2006, we use the formula for percentage growth: \[ \text{Consumption (2006)} = \text{Consumption (2005)} \times \left(1 + \frac{\text{Growth Rate}}{100}\right) \] Plugging in the values:\[ \text{Consumption (2006)} = 91 \times \left(1 + \frac{186.6}{100}\right) = 91 \times 2.866 = 260.726 \text{ million gallons} \] The US consumed approximately 260.7 million gallons in 2006.

Step 2: Calculate 2007 Consumption

In 2006, the US consumed approximately 260.7 million gallons. The growth rate in 2007 was 37.2%. We apply the same percentage growth formula:\[ \text{Consumption (2007)} = \text{Consumption (2006)} \times \left(1 + \frac{\text{Growth Rate}}{100}\right) \] Plugging in the values:\[ \text{Consumption (2007)} = 260.7 \times \left(1 + \frac{37.2}{100}\right) = 260.7 \times 1.372 = 357.1644 \text{ million gallons} \] The US consumed approximately 357.2 million gallons in 2007.

Step 3: Define Scales and Plot Points for Graph

The consumption values for 2005 to 2009 are now needed. These are 91 million gallons for 2005, 260.7 for 2006, and 357.2 for 2007 as calculated. We will continue the pattern to find approximate consumption for 2008 and 2009 using their respective growth rates.For 2008:\[ \text{Consumption (2008)} = 357.2 \times (1 - 0.117) = 315.2 \text{ million gallons}\] For 2009:\[ \text{Consumption (2009)} = 315.2 \times (1 + 0.073) = 338.6 \text{ million gallons}\] Now plot these values on a graph with years 2005 to 2009 on the horizontal axis and gallons consumed (in millions) on the vertical axis.

Key Concepts

Biodiesel Consumption

Biodiesel is a renewable and biodegradable fuel derived from natural resources such as vegetable oils, animal fats, and recycled restaurant greases. This alternative fuel holds great promise in reducing dependency on fossil fuels and minimizing environmental impacts. Biodiesel consumption has gained traction globally as several countries strive towards more sustainable energy sources.

The exercise outlines how the US consumption of biodiesel has varied over the years. Starting with a baseline of 91 million gallons in 2005, the consumption fluctuated significantly due to changing growth rates each year.

Understanding biodiesel consumption patterns is crucial in determining a country's shift towards renewable energy. It helps assess the efficiency and uptake of alternative fuels like biodiesel and predicts future trends in sustainable energy use.

Percentage Growth Calculation

Calculating percentage growth is an essential mathematical concept used to determine how much a quantity has changed over time relative to its initial value. It is particularly useful when analyzing trends like fuel consumption or economic growth.

The formula for obtaining the new value after growth is given by:

• \[\text{New Value} = \text{Original Value} \times \left(1 + \frac{\text{Growth Rate in percentage}}{100}\right)\]

To apply this to biodiesel, consider the growth from 2005 to 2006:

• Given a 186.6% increase, \(\text{New Value} = 91 \times \left(1 + \frac{186.6}{100}\right)\), which results in approximately 260.7 million gallons.

Percentage growth calculations are powerful in projecting future values and can be extended to other contexts like economics and population dynamics.

Graphing Data Points

Graphing data points is an effective way to visually represent information such as biodiesel consumption over several years. Graphs convert numerical data into an easy-to-understand format that can reveal trends and patterns.

When plotting biodiesel consumption, it's important to choose appropriate scales on both the horizontal and vertical axes to ensure clarity. With our data:

• The horizontal axis represents the years from 2005 to 2009,
• The vertical axis should accommodate the range of consumption values, from 91 million gallons to around 338.6 million gallons.

Through the plotted data, a clear upward trend until 2007 followed by slight fluctuations can be seen. Graphs not only present the data succinctly but also help in making informed predictions about future consumption.

Renewable Fuel Sources

Renewable fuel sources like biodiesel are increasingly important in mitigating climate change and reducing reliance on non-renewable energy. Biodiesel produces fewer emissions compared to traditional fossil fuels and can be easily integrated into existing diesel engines without modifications.

Biodiesel is derived from sustainable materials such as:

• Vegetable oils from crops,
• Animal fats,
• Algae,
• Recycled restaurant oils.

These resources renew naturally and provide a sustainable loop for energy production, unlike finite fossil reserves.

By embracing renewable fuel sources, societies can aim for energy independence, reduce greenhouse gas emissions, and promote economic growth through new industries and technologies.