/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 The circulation \(C\) of a certa... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 17

The circulation \(C\) of a certain magazine as a function of time \(t\) is given by the formula $$ C=\frac{5.2}{0.1+0.3^{t}} $$ Here \(C\) is measured in thousands, and \(t\) is measured in years since the beginning of 1992 , when the magazine was started. a. Make a graph of \(C\) versus \(t\) covering the first 6 years of the magazine's existence. b. Express using functional notation the circulation of the magazine 18 months after it was started, and then find that value. c. Over what time interval is the graph of \(C\) concave up? Explain your answer in practical terms. d. At what time was the circulation increasing the fastest? e. Determine the limiting value for \(C\). Explain your answer in practical terms.

Short Answer

Expert verified
Graph first 6 years of C. At 1.5 years, C = 8.93. Concave up until C''(t) > 0. Fastest at maximum C'(t). Limit C = 52.

Step by step solution

01

Graph of C versus t

First, identify that you need to plot the function \( C(t) = \frac{5.2}{0.1 + 0.3^{t}} \). Use a calculator or a graphing tool to compute values of \( C \) for \( t \) ranging from 0 to 6, which corresponds to the first six years. You should begin by calculating a few key points such as \( t = 0, 1, 2, 3, 4, 5, 6 \). Connect these points to form the graph.
02

Functional Notation and Finding Values

For 18 months, \( t \) equals \( 1.5 \) years. Substitute \( t = 1.5 \) into the function: \[ C(1.5) = \frac{5.2}{0.1 + 0.3^{1.5}} \]Calculate \( 0.3^{1.5} \) and substitute back to find \( C(1.5) \). After solving, you determine \( C(1.5) \).
03

Concavity of C

To determine when the graph is concave up, find the second derivative \( C''(t) \). Compute both the first and second derivatives using analytical calculus. Observe where \( C''(t) > 0 \), indicating intervals of concave-up behavior. This requires solving the second derivative and determining critical points.
04

Fastest Increasing Circulation

The rate of change of \( C \) is given by the first derivative \( C'(t) \). To find when circulation is increasing fastest, this involves finding the critical points of \( C'(t) \), specifically where the first derivative is maximized. Solve \( C''(t) = 0 \) to find these points, and verify that they correspond to a maximum.
05

Limiting Value for C

The limiting value of \( C \) as \( t \to \infty \) is the horizontal asymptote of \( C(t) \). Evaluate \( \lim_{t \to \infty} C(t) \). Simplify \( \frac{5.2}{0.1 + 0.3^{t}} \) and find that as \( t \to \infty \), \( 0.3^t \to 0 \). Therefore, \( C \to 52 \).

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Key Concepts

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

Graphing Functions
When graphing a function, such as the circulation formula \( C(t) = \frac{5.2}{0.1 + 0.3^{t}} \), it means plotting points of \( C \) over specific values of \( t \). Here, \( t \) represents time in years. To graph this function, you need:
  • Key points: Choose values for \( t \) such as 0, 1, 2, 3, 4, 5, and 6, representing each year.
  • Calculations: Substitute these \( t \) values into the function to get corresponding \( C \) values.
  • Plotting: Use graph paper or a digital tool to mark these points and connect them to form the curve.

This gives a visual representation of how the magazine circulation changes over time, helping you to easily see any trends.
Concavity of Functions
In mathematics, the concavity of a function tells us about its graphical curvature. For a function \( C(t) \), we find this by calculating the second derivative \( C''(t) \).
  • Concave Up: If \( C''(t) > 0 \), the graph is concave up. It looks like a 'smile'. This is where the rate of change increases.
  • Determining Intervals: By solving \( C''(t) > 0 \), you identify time intervals where the function curves upwards.

In real life, this means the circulation is accelerating during these periods, showing a boost in popularity.
Rate of Change
The rate of change of a function describes how quickly the function's values are increasing or decreasing. For the magazine's circulation \( C(t) \), the first derivative \( C'(t) \) provides this rate.
  • Finding Critical Points: Calculate \( C'(t) \) and solve \( C''(t) = 0 \) to find when the rate of change is highest.
  • Maximum Increase: At these critical points, if \( C'(t) \) is at a maximum, circulation grows fastest.

This concept helps determine when the magazine gained momentum, indicating peak periods of reader interest.
Asymptotic Behavior
Asymptotic behavior in functions refers to how a function behaves as its input becomes very large or very small. For the function \( C(t) \), this is seen as \( t \) approaches infinity.
  • Horizontal Asymptotes: Check \( \lim_{t \to \infty} C(t) \). Here, \( 0.3^t \) tends to zero, simplifying to \( C \to 52 \).
  • Practical Interpretation: This means the circulation stabilizes at a maximum of 52,000 as time progresses.

Understanding this concept is essential for predicting long-term behavior, such as market saturation in this case, showing stability in readership.

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Most popular questions from this chapter

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