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Chapter 4: Problem 8

Reaction time: For certain decisions, the time it takes to respond is a logarithmic function of the number of choices faced. \({ }^{32}\) One model is $$ R=0.17+0.44 \log N, $$ where \(R\) is the reaction time in seconds and \(N\) is the number of choices. a. Draw a graph of \(R\) versus \(N\). Include values of \(N\) from 1 to 10 choices. b. Express using functional notation the reaction time if there are seven choices, and then calculate that time. c. If the reaction time is to be at most \(0.5\) second, how many choices can there be? d. If the number of choices increases by a factor of 10 , what happens to the reaction time? e. Explain in practical terms what the concavity of the graph means.

Short Answer

Expert verified
a. Graph \( R \) from 1 to 10. b. \( R(7) \approx 0.54 \) seconds. c. Up to 2 choices. d. \( R \) increases by \( 0.44 \log 10 \approx 0.44 \).

Step by step solution

01

Graph Setup

To draw the graph of \( R \) versus \( N \), we will calculate \( R \) for each integer \( N \) from 1 to 10 by substituting \( N \) into the equation \( R = 0.17 + 0.44 \log N \).

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Key Concepts

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

Understanding Reaction Time
Reaction time is the duration it takes for an individual to respond to a stimulus. In this exercise, we look at how reaction time changes based on the number of available choices. For decisions involving multiple options, the reaction time can be modeled using a logarithmic function. This means the time taken grows at a decreasing rate as the number of choices increases. In simpler terms, each additional choice adds less to the reaction time compared to the previous one. This is why understanding the logarithmic nature of the function is crucial:
  • The reaction time, denoted as \( R \), is given by the equation \( R = 0.17 + 0.44 \log N \).
  • \( N \) represents the number of choices available.
  • The logarithmic function reflects how people process information鈥攎ore choices do increase time, but not linearly.
Impact of Number of Choices
The number of choices, denoted by \( N \), directly affects the reaction time due to its position in the logarithmic function used in this problem. As \( N \) increases, it impacts the reaction time \( R \) according to the formula \( R = 0.17 + 0.44 \log N \). Let's explore this relationship:
  • For \( N = 1 \), the reaction time \( R \) would be close to 0.17 seconds, since \( \log 1 = 0 \).
  • As \( N \) increases to 10, the reaction time doesn't simply add fixed increments; instead, it increases progressively less per additional choice.
  • This imposes a limit on how quickly we can respond as choices multiply, highlighting practical constraints in decision-making scenarios with many alternatives.
Thus, the number of choices plays a key role in determining how swiftly decisions can be made.
Graph Interpretation
Interpreting the graph of the function \( R = 0.17 + 0.44 \log N \) involves observing how reaction time changes as the number of choices increases. The graph is not a straight line because the functional relationship is logarithmic.
  • The graph starts at \( N = 1 \) with the lowest reaction time and gradually rises as \( N \) increases.
  • The slope decreases as \( N \) becomes larger, illustrating that each additional choice contributes less to the increase in reaction time.
  • The concavity of the graph, bending downwards, indicates diminishing returns in reaction time increase鈥攅ach new choice causes a smaller increase in reaction time than the previous choice.
Understanding this graph is vital for visually grasping the concept of how choices impact reaction speed.
Functional Notation Usage
Functional notation is a way to express relationships between variables in mathematical form. In this problem, it helps us succinctly and clearly see the connection between different inputs (number of choices) and the output (reaction time).
  • The equation \( R(N) = 0.17 + 0.44 \log N \) is written in functional notation, where \( R(N) \) denotes the reaction time function for a given number \( N \) of choices.
  • To find the reaction time for seven choices, use \( R(7) \). The notation makes it easy to calculate for specific values by substitution. \( R(7) = 0.17 + 0.44 \log 7 \) provides the reaction time for this scenario.
  • The simplicity of functional notation allows it to be easily adapted for different scenarios, supporting efficient problem-solving.
This method of expressing relationships is particularly useful in advanced math and science, where clear, concise representation of data is crucial.

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Most popular questions from this chapter

Moore's law: The speed of a computer chip is closely related to the number of transistors on the chip, and the number of transistors on a chip has increased with time in a remarkably consistent way. In fact, in the year 1965 Dr. Gordon E. Moore (now chairman emeritus of Intel Corporation) observed a trend and predicted that it would continue for a time. His observation, now known as Moore's law, is that every two years or so a chip is introduced with double the number of transistors of its fastest predecessor. This law can be restated \({ }^{28}\) in the following way: If time increases by 1 year, then the number of transistors is multiplied by \(10^{0.15}\). More generally, the rule is that if time increases by \(t\) years, then the number of transistors is multiplied by \(10^{0.15 t}\). For example, after 8 years the number of transistors is multiplied by \(10^{0.15 \times 8}\), or about 16 . The Dual-Core Itanium 2 processor was introduced by Intel Corporation in the year \(2006 .\) a. If a chip were introduced in the year 2013 , how many times the transistors of the Dual-Core Itanium 2 would you expect it to have? Round your answer to the nearest whole number. b. The limit of conventional computing will be reached when the size of a transistor is on the scale of an atom. At that point the number of transistors on a chip will be 1000 times that of the Dual-Core Itanium 2 . When, according to Moore's law, will that limit be reached? c. Even for unconventional computing, the laws of physics impose a limit on the speed of computation. \({ }^{29}\) The fastest speed possible corresponds to having about \(10^{40}\) times the number of transistors as on the Dual-Core Itanium 2. Assume that Moore's law will continue to be valid even for unconventional computing, and determine when this limit will be reached. Round your answer to the nearest century.

An exponential model with unit adjustment: Show that the following data are exponential and find a formula for an exponential model. (Note: It will be necessary to make a unit adjustment. For this problem, round your answers to three decimal places.) $$ \begin{array}{|c|c|c|c|} \hline t & g(t) & t & g(t) \\ \hline 0 & 38.30 & 12 & 16.04 \\ \hline 4 & 28.65 & 16 & 11.99 \\ \hline 8 & 21.43 & 20 & 8.97 \\ \hline \end{array} $$

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Atmospheric pressure: The table below gives a measurement of atmospheric pressure, in grams per square centimeter, at the given altitude, in kilometers.17 $$ \begin{array}{|c|c|} \hline \text { Altitude } & \text { Atmospheric pressure } \\ \hline 5 & 569 \\ \hline 10 & 313 \\ \hline 15 & 172 \\ \hline 20 & 95 \\ \hline 25 & 52 \\ \hline \end{array} $$ (For comparison, 1 kilometer is about 0.6 mile, and 1 gram per square centimeter is about 2 pounds per square foot.) a. Plot the data on atmospheric pressure. b. Make an exponential model for the data on atmospheric pressure. c. What is the atmospheric pressure at an altitude of 30 kilometers? d. Find the atmospheric pressure on Earth鈥檚 surface. This is termed standard atmospheric pressure. e. At what altitude is the atmospheric pressure equal to 25% of standard atmospheric pressure?

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