Problem 37 - Applied 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. $$\begin{array}{c|c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 \\ \hline \text { * growth } & -12.5 & 92.9 & 237 & 186.6 & 37.2 & -11.7 & 7.3 \\ \hline \end{array}$$ (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

The US consumed approximately 261 million gallons in 2006 and 358 million gallons in 2007.

Step 1: Understand the Growth Table

The table provides the percentage growth in US biodiesel consumption for each consecutive year starting from 2003. Notably, in 2005, the growth was 237% from the previous year.

Step 2: Use 2005 Consumption as a Baseline

According to the problem, in 2005 the US consumed 91 million gallons of biodiesel. This is our starting point to calculate the consumption for future years.

Step 3: Calculate 2006 Biodiesel Consumption

Using the growth rate for 2006, which is 186.6%, calculate the consumption for 2006.Calculate the increase:\[\text{Increase} = 91 \times \frac{186.6}{100}\]Multiply the base (91 million gallons) by the increase to find 2006 consumption:\[\text{2006 consumption} = 91 + 91 \times \frac{186.6}{100}\approx 91 + 169.806 = 260.806 \text{ million gallons}\]

Step 4: Calculate 2007 Biodiesel Consumption

Using the calculated 2006 consumption, determine the consumption for 2007 with a 37.2% increase:Calculate the increase:\[\text{Increase} = 260.806 \times \frac{37.2}{100}\approx 97.159\]Add this increase to the 2006 consumption:\[\text{2007 consumption} = 260.806 + 97.159 = 357.965 \text{ million gallons}\]

Step 5: Plot the Biodiesel Consumption from 2005 to 2009

Create a set of points to graph the consumption:- 2005: 91 million gallons- 2006: 260.806 million gallons- 2007: 357.965 million gallonsFor a complete graph, further calculate 2008 and 2009 values using respective growth rates:2008: \[\text{Decrease: } 357.965 \times \frac{11.7}{100} = 41.842\]\[\text{2008 consumption} = 357.965 - 41.842 = 316.123 \text{ million gallons}\]2009: \[\text{Increase: } 316.123 \times \frac{7.3}{100} = 23.076\]\[\text{2009 consumption} = 316.123 + 23.076 = 339.199 \text{ million gallons}\]Now plot these data points on a graph with the x-axis representing years (2005 to 2009) and the y-axis representing millions of gallons of biodiesel.

Key Concepts

Percentage Growth

Percentage growth refers to how much a quantity has increased or decreased in relation to its previous value, expressed as a percentage. In our biodiesel example, we use percentage growth to calculate how consumption changes year-on-year. This simple calculation involves multiplying the previous year's consumption by the percentage growth rate.
For instance, with 2005 as the baseline ( 91 million gallons), the growth in 2006 was 186.6%. To calculate this, think of it as adding 186.6% of 2005's total to itself. This results in 260.806 million gallons for 2006.
When computing percentage growth, remember it can be either positive (indicating growth) or negative (indicating decline). This helps to see if biodiesel usage is expanding or contracting annually.

Renewable Resources

Biodiesel is a type of biofuel derived from renewable resources. Unlike fossil fuels, which are extracted from finite sources, renewable resources are naturally replenished on a human timescale.
The primary sources of biodiesel include:

• Food crops, such as soybeans and corn
• Algae, which are fast-growing and can be cultivated in various water bodies
• Animal fats, often a byproduct of meat processing

Using biodiesel supports sustainability efforts because it decreases our reliance on non-renewable energy sources. Moreover, when these organic materials are turned into fuel, they contribute less directly to greenhouse gas emissions than traditional fuels. This positions biodiesel as a cleaner alternative, aligning with global efforts to increase the use of renewable energy sources.

Annual Consumption Graph

An annual consumption graph visually represents how a variable—in this case, biodiesel consumption—changes over a set of time, expressed in millions of gallons. It is a valuable tool to identify trends and patterns across the years.
Such a graph often has years on the x-axis (the horizontal line) and the quantity of biodiesel consumed on the y-axis (the vertical line). By plotting points corresponding to the data from each year, you can create a meaningful visual that depicts the rise and fall of consumption.
For example, starting from 2005 with 91 million gallons, various percentage changes add up to different quantities each year, which can be plotted as points. Linking these points creates a line graph, making it easier to comprehend changes and analyze the impact of certain events or policies on consumption. Once the graph is plotted, trends like increases or decreases are apparent at a glance.