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Chapter 4: Problem 27

The proportion of students in a psychology experiment who could remember an eight-digit number correctly for \(t\) minutes was \(0.9-0.2 \ln t\) (for \(t>1\) ). Find the proportion that remembered the number for 5 minutes.

Short Answer

Expert verified
The proportion is approximately 0.578.

Step by step solution

01

Understand the Given Formula

The proportion of students who remember the number is given by the function \( P(t) = 0.9 - 0.2 \ln t \). Our task is to use this formula to find the proportion at \( t = 5 \) minutes.
02

Substitute the Time Value

Substitute \( t = 5 \) into the formula to find \( P(5) \). So, we have \( P(5) = 0.9 - 0.2 \ln 5 \).
03

Calculate the Natural Logarithm

Calculate the value of \( \ln 5 \). Using a calculator, \( \ln 5 \approx 1.6094 \).
04

Compute the Product

Calculate \( 0.2 \times \ln 5 \). Using the result from Step 3, we have \( 0.2 \times 1.6094 \approx 0.32188 \).
05

Find the Final Proportion

Subtract the result from Step 4 from 0.9: \( 0.9 - 0.32188 = 0.57812 \). Thus, \( P(5) \approx 0.57812 \).

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Key Concepts

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

Exponential Functions
Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are often used to model growth processes, like population growth or compound interest.
They are powerful in modeling situations where quantities grow at a constant rate over time.
In simpler terms, if you start with a quantity, it keeps multiplying by a fixed factor in equal time intervals.

In applications like finance and biology:
  • The base represents the percentage rate of growth or decay.
  • The exponent represents time periods.
When modeling these situations, exponential functions can provide insight into how systems evolve over time.
Understanding exponential functions helps to decode complex scenarios into understandable growth trends.
Natural Logarithms
Natural logarithms are logarithms to the base of Euler's number, approximately 2.718. It is denoted as \( \ln \) and allows us to undo exponential functions.
A natural logarithm answers the question: 'To what power must \( e \) be raised, to obtain a given number?' This is especially useful in multiple fields like science and finance, where exponential growth or decay models are common.

For example, if \( \ln x = y \), it means \( e^y = x \). Let's see some key features of natural logarithms:
  • \( \ln 1 = 0 \) because \( e^0 = 1 \).
  • It is undefined for non-positive numbers.
  • It converts multiplication into addition and exponentiation into multiplication, e.g., \( \ln(AB) = \ln A + \ln B \).
In calculations, natural logarithms help simplify complex exponential expressions, making them essential in calculus and many applied sciences.
Mathematical Modeling
Mathematical modeling uses mathematical expressions to represent real-world scenarios. This helps simulate complex systems, predict future behaviors, and understand underlying patterns.
Models use equations, like the exponential decay function in the psychology experiment example, to approximate how systems work in reality.

Characteristics of effective mathematical models include:
  • Accuracy: How well the model predicts or describes outcomes.
  • Simplicity: The model shouldn't be unnecessarily complex.
  • Generality: A good model applies to a wide variety of scenarios.
Mathematical modeling requires balancing simplicity and realism to ensure the model is both practical and accurate.
Through these models, you can experiment with different variables to understand effects and optimize outcomes in fields like economics, engineering, and ecology.

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Most popular questions from this chapter

\(55-58\). For each function: a. Find \(f^{\prime}(x)\) b. Evaluate the given expression and approximate it to three decimal places. $$ f(x)=\ln \left(e^{x}-1\right), \text { find and approximate } f^{\prime}(3) . $$

PERSONAL FINANCE: Earnings and Calculus A study found that one's earnings are affected by the mathematics courses one has taken. In particular, compared to someone making \(\$ 40,000\) who had taken no calculus, a comparable person who had taken \(x\) years of calculus would be earning \(\$ 40,000 e^{0.195 x}\). Find the rate of change of this function at \(x=1\) and interpret vour answer.

101-107. Choose the correct answer: $$ \begin{aligned} &\frac{d}{d x} e^{f(x)}=\quad \text { a. } e^{f(x)} f^{\prime}(x)\\\ &\text { b. } e^{f^{\prime}(x)} f(x) \text { c. } e^{f^{\prime}(x)} f^{\prime}(x) \end{aligned} $$

A company is considering two ways to depreciate a truck: straight line or by a fixed percentage. If each way begins with a value of \(\$ 25,000\) and ends 5 years later with a value of \(\$ 1000\), which way will result in the bigger change in value during the first year? During the last year?

ATHLETICS: How Fast Do Old Men Slow Down? The fastest times for the marathon \((26.2\) miles \()\) for male runners aged 35 to 80 are approximated by the function $$ f(x)=\left\\{\begin{array}{ll} 106.2 e^{0.0063 x} & \text { if } x \leq 58.2 \\ 850.4 e^{0.000614 x^{2}-0.0652 x} & \text { if } x>58.2 \end{array}\right. $$ in minutes, where \(x\) is the age of the runner. Source: Review of Economics and Statistics LXXVI a. Graph this function on the window [35,80] by [0,240] [Hint: On some graphing calculators, $$ \text { enter } y_{1}=\left(106.2 e^{0.0063 x}\right)(x \leq 58.2)+ $$ \(\left.\left(850.4 e^{0.000614 x^{2}-0.0652 x}\right)(x>58.2) .\right]\) b. Find \(f(35)\) and \(f^{\prime}(35)\) and interpret these numbers. [Hint: Use NDERIV or \(d y / d x\). c. Find \(f(80)\) and \(f^{\prime}(80)\) and interpret these numbers.
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