/*! 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 78 A rumor is spread in a school. F... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 78

A rumor is spread in a school. For \(00\) the time \(t\) at which a fraction \(p\) of the school population has heard the rumor is given by $$t(p)=\int_{a}^{p} \frac{b}{x(1-x)} d x$$ (a) Evaluate the integral to find an explicit formula for \(t(p) .\) Write your answer so it has only one ln term. (b) At time \(t=0\) one percent of the school population \((p=0.01)\) has heard the rumor. What is \(a ?\) (c) At time \(t=1\) half the school population \((p=0.5)\) has heard the rumor. What is \(b ?\) (d) At what time has \(90 \%\) of the school population \((p=0.9)\) heard the rumor?

Short Answer

Expert verified
(a) \(t(p) = b \ln \frac{p(1-a)}{a(1-p)}\); (b) \(a = 0.01\); (c) \(b = \frac{1}{\ln 99}\); (d) Calculate \(t(0.9)\).

Step by step solution

01

Evaluate the integral

We start by evaluating the integral \(\int_{a}^{p} \frac{b}{x(1-x)} \, dx\) given in the problem:\[t(p) = \int_{a}^{p} \frac{b}{x(1-x)} \, dx = b \left(\int_{a}^{p} \frac{1}{x} \, dx + \int_{a}^{p} \frac{1}{1-x} \, dx \right)\]The results of these integrals are:1. \(\int \frac{1}{x} \, dx = \ln |x|\)2. \(\int \frac{1}{1-x} \, dx = -\ln |1-x|\)Combine these to find an explicit expression:\[b \left(\ln |p| - \ln |1-p| - (\ln |a| - \ln |1-a|)\right)\]By simplifying, we use the property \(\ln \frac{m}{n} = \ln m - \ln n\), obtaining:\[t(p) = b \left(\ln \frac{p(1-a)}{a(1-p)}\right)\]
02

Determine the value of \(a\)

Given that at \(t = 0\), one percent of the population \((p = 0.01)\) has heard the rumor, we set up the equation using \(t(p)\):\[t(0.01) = b \left(\ln \frac{0.01(1-a)}{a(0.99)}\right) = 0\]Since \(b > 0\), this implies:\[\frac{0.01(1-a)}{a(0.99)} = 1\]Solving \(0.01(1-a) = a(0.99)\):\[0.01 - 0.01a = 0.99a\]\[0.01 = 1a\]Thus, \(a = 0.01\).
03

Determine the value of \(b\)

At time \(t = 1\), half the school population \((p = 0.5)\) has heard the rumor, so we substitute into our expression for \(t(p)\):\[t(0.5) = b \left(\ln \frac{0.5(1-0.01)}{0.01(0.5)}\right) = 1\]Evaluate the expression inside the logarithm:\[\ln \frac{0.5 \times 0.99}{0.01 \times 0.5} = \ln 99\]Thus:\[b \ln 99 = 1\]So, \(b = \frac{1}{\ln 99}\).
04

Find the time for 90% coverage

We need to find the time \(t\) when \(90\%\) of the population \((p = 0.9)\) has heard the rumor:\[t(0.9) = b \left(\ln \frac{0.9(1-0.01)}{0.01(0.1)}\right)\]Calculate the expression inside the logarithm:\[\ln \frac{0.9 \times 0.99}{0.01 \times 0.1} = \ln 891\]Substitute \(b = \frac{1}{\ln 99}\):\[t(0.9) = \frac{1}{\ln 99} \times \ln 891\]Thus, calculate \(t(0.9)\) numerically.

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Key Concepts

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

Integration Techniques
Understanding integration techniques is vital in calculus, especially for solving problems involving exponential and logarithmic functions. Integration techniques offer ways to simplify and solve complex integrals. The key technique used in this problem is the method of partial fraction decomposition.

Here's a breakdown of the process:
  • The given integral is: \(\int \frac{b}{x(1-x)} \, \mathrm{d}x\).
  • This can be split into two separate integrals using partial fractions, leading to: \(\int \frac{1}{x} \, \mathrm{d}x + \int \frac{1}{1-x} \, \mathrm{d}x\).
  • By evaluating these integrals, you find: \(\ln |x|\) and \(-\ln |1-x|\).
The solution involves combining these results and applying logarithmic properties to simplify and arrive at the final answer. This method clearly demonstrates how to handle integrals involving rational functions.
Exponential Growth Models
Exponential growth models are used to describe how quantities grow rapidly over time. In our problem, the spread of a rumor illustrates exponential growth. At its core, exponential growth means that a quantity increases at a rate proportional to its current size.

Consider the elements of the problem:
  • The fraction \(p\) represents the proportion of the population aware of the rumor. This value is increasing over time \(t\).
  • The mathematical model considers how the rumor spreads gradually through the population, with the rate of spread diminishing as more people are aware.
Exponentially growing systems like this are common in real-world phenomena, including populations, investments, and technologies, all driven by similar mathematical principles.
Logarithmic Functions
Logarithmic functions are essential for simplifying expressions and solving equations involving exponential growth. They allow us to express exponential relationships in a linear form, which can make analysis much easier.

In this exercise:
  • The logarithmic property \(\ln \frac{m}{n} = \ln m - \ln n\) is used to combine terms efficiently.
  • This property enables the solution to express the time \(t\) as a single logarithm, simplifying further calculations.
Understanding logarithms helps reveal patterns in growth data and simplify complex equations. In our context, they transform the problem of exponential spreading into manageable algebraic steps.

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Most popular questions from this chapter

The gamma function is defined for all \(x>0\) by the rule $$\Gamma(x)=\int_{0}^{\infty} t^{x-1} e^{-t} d t$$ (a) Find \(\Gamma(1)\) and \(\Gamma(2)\) (b) Integrate by parts with respect to \(t\) to show that, for positive \(n\) $$\Gamma(n+1)=n \Gamma(n)$$ (c) Find a simple expression for \(\Gamma(n)\) for positive integers \(n\)

The moment-generating function, \(m(t),\) which gives useful information about the normal distribution of statistics, is defined by $$m(t)=\int_{-\infty}^{\infty} e^{t x} \frac{e^{-x^{2} / 2}}{\sqrt{2 \pi}} d x$$ Find a formula for \(m(t) .\) [Hint: Complete the square and use the fact that \(\left.\int_{-\infty}^{\infty} e^{-x^{2} / 2} d x=\sqrt{2 \pi} .\right]\)

Decide whether the statements are true or false. Give an explanation for your answer. When integrating by parts, it does not matter which factor we choose for \(u.\)

Explain what is wrong with the statement. $$\int \cos \left(x^{2}\right) d x=\sin \left(x^{2}\right) /(2 x)+C$$

If appropriate, evaluate the following integrals by substitution. If substitution is not appropriate, say so, and do not evaluate. (a) \(\int x \sin \left(x^{2}\right) d x\) (b) \(\int x^{2} \sin x d x\) (c) \(\int \frac{x^{2}}{1+x^{2}} d x\) (d) \(\int \frac{x}{\left(1+x^{2}\right)^{2}} d x\) (e) \(\int x^{3} e^{x^{2}} d x\) (f) \(\int \frac{\sin x}{2+\cos x} d x\)
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