Problem 45 - Calculus

Question

Hydroelectric power is electric power generated by the force of moving water. The table shows the annual percent change in hydroelectric power consumption by the US industrial sector. 15 $$\begin{array}{c|c|c|c|c|c} \hline \text { Year } & 2005 & 2006 & 2007 & 2008 & 2009 \\ \hline \% \text { growth over previous yr } & -1.9 & -10 & -45.4 & 5.1 & 11 \\\ \hline \end{array}$$ (a) According to the US Department of Energy, the US industrial sector consumed about 29 trillion BTUs of hydroelectric power in \(2006 .\) Approximately how much hydroelectric power (in trillion BTUs) did the US consume in \(2007 ?\) In \(2005 ?\) (b) Graph the points showing the annual US consumption of hydroelectric power, in trillion BTUs, for the years 2004 to \(2009 .\) Label the scales on the horizontal and vertical axes. (c) According to this data, when did the largest yearly decrease, in trillion BTUs, in the US consumption of hydroelectric power occur? What was this decrease?

Step-by-Step Solution

Answer

(a) 2007: 15.834 TBtu, 2005: 29.570 TBtu. (b) Plot points for 2004-2009. (c) Largest decrease: 2006 to 2007, 13.166 TBtu.

Step 1: Calculate hydropower consumption for 2007

The consumption in 2006 was 29 trillion BTUs, and the percent change from 2006 to 2007 was -45.4%. Calculate 2007 consumption using: \[\text{Consumption in 2007} = \text{Consumption in 2006} \times \left(1 - \frac{45.4}{100}\right)\]\[= 29 \times (1 - 0.454) = 29 \times 0.546 = 15.834 \text{ trillion BTUs}\]

Step 2: Calculate hydropower consumption for 2005

The consumption in 2006 was 29 trillion BTUs, and the percent change from 2005 to 2006 was -1.9%. Find 2005 consumption using the inverse method:\[\text{Consumption in 2005} = \frac{\text{Consumption in 2006}}{1 - \frac{1.9}{100}}\]\[= \frac{29}{0.981} = 29.570 \text{ trillion BTUs}\]

Step 3: Estimate hydropower consumption for other years

For 2008, the percent change was 5.1%, so calculate:\[= 15.834 \times (1 + 0.051) = 16.640 \text{ trillion BTUs}\]For 2009, the percent change was 11%, so calculate:\[= 16.640 \times (1 + 0.11) = 18.470 \text{ trillion BTUs}\]

Step 4: Estimate hydropower consumption for 2004

Starting from the 2005 result, the percent change was -1.9%, so to get 2004:\[\text{Consumption in 2004} = \frac{29.570}{1 - \frac{1.9}{100}} = \frac{29.570}{0.981} = 30.138 \text{ trillion BTUs}\]

Step 5: Graph the data

Plot the following points on a graph with the years on the horizontal axis and consumption in trillion BTUs on the vertical axis: - 2004: 30.138 - 2005: 29.570 - 2006: 29 - 2007: 15.834 - 2008: 16.640 - 2009: 18.470

Step 6: Calculate the largest yearly decrease

Compare the consumption differences: - 2005 to 2006: 29.570 - 29 = 0.570 - 2006 to 2007: 29 - 15.834 = 13.166 - 2007 to 2008: 15.834 - 16.640 = -0.806 (increase) - 2008 to 2009: 16.640 - 18.470 = -1.830 (increase) The largest decrease was from 2006 to 2007, at 13.166 trillion BTUs.

Key Concepts

Percent Change Calculation

The percent change calculation is a foundational concept often used to measure how much a value has increased or decreased compared to its previous state. This is particularly useful for comparing data over different years.

To compute the percent change, you need the original figure and how much it has changed. The formula is:

\[\text{Percent Change} = \left( \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \right) \times 100\]For example, if the data for hydroelectric consumption decreased from 29 trillion BTUs in 2006 to 15.834 trillion BTUs in 2007, we apply the percent change formula to find out how significant that drop is in percentage terms.

By substituting the values into the formula, we find a percent change of -45.4%, confirming the data provided. This shows how each yearly consumption relates to the previous year's figure through these changing percentages.

Graphing Data

Graphing data is a visual method to present numerical information over a period. It helps in easily identifying trends, patterns, and outliers in the data set. When graphing the consumption of hydroelectric power over multiple years:

• The horizontal axis (x-axis) is typically set to represent time, such as years (e.g., 2004, 2005, etc.).
• The vertical axis (y-axis) shows the value being measured—in this case, hydroelectric consumption in trillion BTUs.

Step-by-step, you plot each data point corresponding to each year's power consumption. For instance, you would plot 30.138 trillion BTUs for 2004, 29.570 trillion BTUs for 2005, and so forth.

Each point plotted can be connected to form a line graph, visually highlighting how consumption levels changed over the span of those years.

Graphing allows for quick assessment of which year experienced the most significant shift and helps validate calculations of percent changes by visual representation.

Yearly Decrease Analysis

Understanding yearly decreases is crucial to identify particular years of significant drop in values. When analyzing data, especially consumption figures, identifying the largest yearly decrease is an essential step.

For hydroelectric power consumption, this involves comparing the BTU figures year-on-year to isolate when the consumption decreased the most. For example, in our data set:

• Between 2006 and 2007, consumption dropped from 29 trillion BTUs to approximately 15.834 trillion BTUs. The difference, 13.166 trillion BTUs, is calculated and represents the actual decrease between these two years.
• Comparing this difference with others, you'll notice it's significantly larger, thus marking 2006 to 2007 as the year with the largest drop.

Examining these annual changes not only identifies specific years of interest but also helps assess the impact of external factors, such as policy shifts or climatic conditions, that may have influenced the consumption. Keeping track of such insights can guide future decisions in energy management.