Problem 44 - 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) True or false: The annual US consumption of biodiesel grew exponentially from 2003 to 2005 Justify your answer without doing any calculations. (b) According to this data, during what single year(s), if any, did the US consumption of biodiesel at least double? (c) According to this data, during what single year(s), if any, did the US consumption of biodiesel at least triple?

Step-by-Step Solution

Answer

(a) False; (b) 2005 and 2006; (c) 2005.

Step 1: Understand Exponential Growth

Exponential growth occurs when a quantity increases by a consistent percentage over multiple periods. To determine exponential growth from 2003 to 2005, we examine whether there is a consistent percentage increase over these years.

Step 2: Examine Growth Percentages from 2003 to 2005

Observe the growth percentages for the years 2003, 2004, and 2005. The values are -12.5% for 2003, 92.9% for 2004, and 237% for 2005.

Step 3: Determine Exponential Growth (2003-2005)

In 2003, consumption decreased. In 2004 and 2005, the growth percentages are large but not consistent. Thus, the growth is not exponential, as there is no consistent percentage increase.

Step 4: Identify Doubling (100% Growth) in Single Year

To find when consumption at least doubled, look for 100% or more growth in a single year. The data shows 237% in 2005 and 186.6% in 2006. Both years exceed 100%, indicating consumption at least doubled in these years.

Step 5: Identify Tripling (200% Growth) in Single Year

To find when consumption at least tripled, look for 200% or more growth in a single year. 2005 shows a 237% increase, indicating consumption more than tripled in this year.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a constant percentage rate over a set period. Unlike linear growth, where the increase is by a fixed amount, exponential growth leads to larger and larger increments as time progresses.

This growth pattern is often seen in populations, investments, and certain types of resource consumption. In the context of US biodiesel consumption, if the growth percentage remains steady across multiple years without any significant fluctuations, it indicates exponential growth.

• A consistent percentage increase each year.
• Leads to rapid increases in total consumption.
• Results in graph curves that become steeper over time.

However, the data from 2003 to 2005 shows significant fluctuation, with an initial negative growth followed by varying positive growth percentages. This inconsistency means the growth isn’t truly exponential over this period.

US Biodiesel Consumption

The consumption of biodiesel in the US has evolved over the years with varying growth patterns. Biodiesel, a renewable energy source, is extracted from natural resources like vegetable oils and animal fats.

Understanding such growth can help policymakers and companies plan for future energy needs and infrastructures. The data from 2003 to 2009 presents a unique insight into how biodiesel consumption has changed over time in the US.

• Started with a decline in 2003.
• Significant rise in 2004 and 2005, hinting at increased adoption and production.
• Fluctuating growth percentages reflect economic, technological, and policy changes.

It’s crucial for students to understand these trends to appreciate the challenges and opportunities in renewable energy sectors.

Growth Percentage Analysis

Analyzing growth percentages allows us to understand how quickly or slowly a particular metric is changing. In this exercise, we analyze biodiesel consumption percentages for insights.

Each percentage indicates how much more biodiesel was consumed compared to the previous year. For significant growth analysis:

• Negative percentages indicate a decrease in consumption.
• Positive percentages suggest growth, with larger numbers representing more rapid increases.
• Analyzing these can help determine when the most significant changes occurred.

This kind of analysis is critical for identifying the years with the most impact, like 2005's 237% increase, marking a significant uptake in biodiesel consumption.

Doubling and Tripling Growth

Doubling and tripling refer to the increase in a quantity to twice or thrice its original size. In growth percentage terms, a 100% increase means doubling, and a 200% increase signifies tripling.

Examining such growth in the context of US biodiesel consumption:

• 2004 did not see doubling or tripling, as growth only reached 92.9%.
• 2005 showed a dramatic 237% increase, more than tripling consumption.
• In 2006, a 186.6% growth indicated consumption more than doubled, but not tripled again.

Understanding these large jumps is essential for grasping periods of rapid expansion or adoption of technologies, indicating shifts in energy policies or market demands.