Question

The table shows the amounts \(A\) (in billions of dollars per year) spent on \(\mathrm{R\&D}\) in the United States from 1980 through 2004 , where \(t\) is the year, with \(t=0\) corresponding to 1980 . Approximate the average rate of change of \(A\) during each period. (a) \(1980-1985\) (b) \(1985-1990\) (c) \(1990-1995\) (d) \(1995-2000\) (e) \(1980-2004\) (f) \(1990-2004\) $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline A & 63 & 72 & 81 & 90 & 102 & 115 & 120 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline A & 126 & 134 & 142 & 152 & 161 & 165 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline A & 166 & 169 & 184 & 197 & 212 & 228 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 19 & 20 & 21 & 22 & 23 & 24 \\ \hline A & 245 & 267 & 277 & 276 & 292 & 312 \\ \hline \end{array} $$

Short answer

The calculated average rates of change (in billion dollars per year) are: (a) 10.4, (b) 12.2, (c) 12.6, (d) 10.6, (e) 9.4, and (f) 10.3.

Step-by-step solution

Identify Initial and Final Amounts

For each defined time period, locate and note the corresponding amounts from the given tables. For instance, for the time period 1980 to 1985, the corresponding \(t\) values are 0 and 5 the associated \(A\) values are 63 billion and 115 billion dollars, respectively.

Apply the Formula for Average Rate of Change

Calculate the average rate of change by subtracting the initial value from the final value and dividing by the number of years elapsed. The formula for average rate of change is \[(\text{final value} - \text{initial value}) / (\text{final year} - \text{initial year})\]. Applying this to the first period yields [(115 - 63) / (5 - 0)] = 10.4 billion dollars per year.

Repeat for Remaining Time Periods

Repeat the process for each specified time period: (b) 1985-1990, (c) 1990-1995, (d) 1995-2000, (e) 1980-2004, and (f) 1990-2004. Always using the relevant \(A\) and \(t\) values to calculate the average rate of change in each case.

Key concepts

R&D Expenditure Analysis

Research and Development (R&D) expenditure is a critical factor in driving innovation and long-term economic growth. Analysis of R&D spending trends can provide insights into a country's focus on technological advancement and its potential future competitiveness in the global marketplace. When we look at R&D expenditure data, we aim to understand not only the total amounts invested but also how these investments change over time.

By examining the data from the United States between 1980 and 2004, we can observe the shifts in financial commitment to R&D. Analyzing periods such as 1980-1985, or 1995-2000, helps to reveal the pace at which investment has risen or fallen. The exercise focuses on extracting such information by calculating the average rate of change across different time spans, providing a quantitative measure of growth or decline in R&D expenditures.

Algebraic Formulas

Algebraic formulas are equations that represent relationships between different variables. These formulas are the cornerstone of finding solutions to a wide array of problems in mathematics and its applications in various fields. In our R&D expenditure example, we use a specific algebraic formula to calculate the average rate of change.

The formula itself is expressed as \[(\text{final value} - \text{initial value}) / (\text{final year} - \text{initial year})\]. This kind of formula is a simple yet powerful tool as it allows us to distill complicated datasets into comprehensible outcomes like trends over time. When grasping the use of algebraic formulas, it is essential to understand each component of the formula and how they relate to the problem at hand. With consistent practice, the application of these formulas can move from a mechanical process to an intuitive one.

Rate of Change Calculation

The rate of change calculation is a fundamental concept in mathematics and economics that measures how a quantity changes over time. It's the 'speed' at which one variable changes with respect to another. In the context of R&D expenditures, we calculate the average rate of change to track the variation of spending over a series of years.

Using the table with R&D data, we isolate the periods in question and utilize the average rate of change formula, which takes the difference in expenditure over the period (final minus initial) and divides by the time elapsed. This figure represents the average increase (or decrease if the result is negative) in billions of dollars per year for the given period. Recognizing the calculations for various periods allows us to compare rates of investment over time and develop insights into the strategic direction of R&D initiatives.

Understanding the calculation of the rate of change is not just about applying formulae; it's about reading the story the numbers tell us about past decisions and possibly projecting future investment trends in R&D.