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Chapter 1: Problem 22

Defense spending: Data about recent federal defense spending are given in the accompanying Statistical Abstract of the United States table. Here \(t\) denotes the time, in years, since 1985 and \(D\) denotes federal defense spending, in billions of dollars. $$ \begin{array}{|c|c|} \hline t=\text { Years } & D=\text { Spending } \\ \text { since } 1985 & \text { (billions of dollars) } \\ \hline 0 & 279.0 \\ \hline 5 & 328.4 \\ \hline 10 & 310.0 \\ \hline 15 & 341.6 \\ \hline 20 & 565.5 \\ \hline \end{array} $$ a. Calculate the average yearly rate of change in defense spending from 1990 to \(1995 .\) b. Use your answer from part a to estimate \(D(8)\), and explain what it means. c. Calculate the average yearly rate of change in defense spending from 2000 to 2005 . d. Use your answer from part c to estimate the value of \(D(22)\).

Short Answer

Expert verified
a. -3.68 billion dollars/year; b. 317.36 billion dollars in 1993; c. 44.78 billion dollars/year; d. 655.06 billion dollars in 2007.

Step by step solution

01

Understand the Problem

We need to find the average yearly rate of change in defense spending between specific years, which means calculating the slope of the line between two points on a graph of the given data points.
02

Find the Average Yearly Rate of Change from 1990 to 1995

Using the points for the years 1990 (\(t = 5\)) and 1995 (\(t = 10\)), with spending values \(328.4\) and \(310.0\) respectively, calculate the slope: \[\text{Average Rate} = \frac{D(10) - D(5)}{10 - 5} = \frac{310.0 - 328.4}{10 - 5} = \frac{-18.4}{5} = -3.68 \, ext{billion dollars per year}\].
03

Estimate Defense Spending for Year 1993 (D(8))

Using the rate of change \(-3.68\) calculated, starting from 1990 \(D(5) = 328.4\), estimate the spending for the year 1993 \(t = 8\): \[ D(8) = 328.4 + (-3.68) \times (8 - 5) = 328.4 - 11.04 = 317.36 \, ext{billion dollars} \]. This means that in 1993, the defense spending is estimated to be 317.36 billion dollars.
04

Find the Average Yearly Rate of Change from 2000 to 2005

Using the points for the years 2000 (\(t = 15\)) and 2005 (\(t = 20\)), with spending values \(341.6\) and \(565.5\) respectively, calculate the slope: \[\text{Average Rate} = \frac{D(20) - D(15)}{20 - 15} = \frac{565.5 - 341.6}{20 - 15} = \frac{223.9}{5} = 44.78 \, ext{billion dollars per year}\].
05

Estimate Defense Spending for Year 2007 (D(22))

Using the rate of change \(44.78\) from the previous step, starting from 2005 \(D(20) = 565.5\), estimate the spending for the year 2007 \(t = 22\): \[D(22) = 565.5 + 44.78 \times (22 - 20) = 565.5 + 89.56 = 655.06 \, ext{billion dollars}\]. This indicates that in 2007, the defense spending is estimated to be 655.06 billion dollars.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Defense Spending
Defense spending represents a significant portion of federal budgets and often reflects the government's priorities towards military and security. Tracking and analyzing defense spending over time can reveal trends, shifts in policy, or responses to international events. In this problem, spending is recorded in billions of dollars, starting from the year 1985.
Understanding these historical values helps us put into perspective how expenses on defense have evolved over decades. Often, defense spending is impacted by economic conditions, geopolitical tensions, or technological advancements.
  • It's crucial to recognize that while some years depict a rise in defense spending, others may show a decline, which might indicate strategic reallocations or budget constraints.
  • Comparing these amounts can provide insight into how priorities may shift over time.
Tracking the average rate of change in such data involves understanding and calculating using concepts like those explored in this problem.
Linear Interpolation
Linear interpolation is a process used to estimate values between two known points on a line. It is a simple yet powerful tool, especially useful when dealing with limited data sets. Here, it allows us to predict spending for a year where data isn't explicitly provided by using surrounding data points.
To perform linear interpolation, follow these steps:
  • Identify the two data points that surround the unknown value.
  • Calculate the average rate of change (or slope) between those two known points.
  • Apply that rate to extrapolate or interpolate the desired value, as reflected in the problem when finding defense spending for unspecific years.
Use interpolation cautiously, as it assumes a straight-line relationship, which may not always accurately represent real-world data behaviors.
Data Analysis
Data analysis is the process of systematically applying statistical and logical techniques to describe, illustrate, and evaluate data. In the context of this exercise, analyzing the defense spending data involves calculating average rates of change and making estimations.
Effective data analysis gives us the ability to:
  • Identify trends over specified intervals via calculated slopes.
  • Make informed forecasts for future spending or projections.
  • Detect unusual patterns or anomalies that necessitate deeper investigation.
Data analysis often begins with understanding the problem, gathering relevant data, calculating necessary metrics, and interpreting these metrics within the context, such as federal budget reviews or policymaking decisions.
Statistical Abstract
A Statistical Abstract is a comprehensive collection of statistics covering various aspects of a particular topic, often released annually. It serves as a valuable resource for obtaining reliable and comprehensive data referencing specific themes, like economic indicators or government spending.
For defense spending, statistical abstracts can provide historical context, compare spending relative to other sectors, and aid in detailed year-by-year analysis.
  • These abstracts encapsulate data from numerous sources, ensuring a thorough aggregation of statistics.
  • They assist policymakers, researchers, and analysts by providing foundational data for further exploration and decision-making.
In this exercise, the Statistical Abstract of the United States serves as the foundation from which data is extracted to ascertain average rates of change and perform interpolations.

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