Problem 83 - Contemporary Precalculus

Question

One person with a flu virus visited the campus. The number \(T\) of days it took for the virus to infect \(x\) people was given by: $$ T=-.93 \ln \left[\frac{7000-x}{6999 x}\right] $$ (a) How many days did it take for 6000 people to become infected? (b) After two weeks, how many people were infected?

Step-by-Step Solution

Answer

**Answer:** It takes about 5.516 days for 6000 people to become infected, and after two weeks, approximately 3125 people are infected.

Step 1: (a) Finding the number of days to infect 6000 people

We are given that x=6000. Substituting this value into the equation, we get: $$ T=-0.93 \ln \left[\frac{7000-6000}{6999 \cdot 6000}\right] $$ Now, we solve for T: $$ T=-0.93 \ln \left[\frac{1000}{6999 \cdot 6000}\right] $$ Use a calculator or software to compute the natural logarithm and then multiply by -0.93: $$ T \approx 5.516 $$ So, it took about 5.516 days for 6000 people to become infected.

Step 2: (b) Finding the number of people infected after two weeks

We are given that T=14 days. Substituting this value into the equation, we get the following equation in terms of x: $$ 14=-0.93 \ln \left[\frac{7000-x}{6999 \cdot x}\right] $$ Now, we would like to solve for x. First, divide both sides of the equation by -0.93: $$ \frac{14}{-0.93} = \ln \left[\frac{7000-x}{6999 \cdot x}\right] $$ Now, let's take the exponent by the base e to both sides: $$ e^{\frac{14}{-0.93}} = \frac{7000-x}{6999 \cdot x} $$ Now, we multiply both sides by the denominator to get rid of the fraction: $$ 6999xe^{\frac{14}{-0.93}} = 7000-x $$ Now, we would like to make x the subject of the equation. Move -x to the left side and 6999xe^{\frac{14}{-0.93}} to the right side: $$ x+ 6999xe^{\frac{14}{-0.93}}= 7000 $$ Factor out x from the left-hand side: $$ x(1 + 6999e^{\frac{14}{-0.93}}) = 7000 $$ Now, divide both sides by the factor in parentheses: $$ x = \frac{7000}{1 + 6999e^{\frac{14}{-0.93}}} $$ Use a calculator or software to evaluate the expression: $$ x \approx 3125.339 $$ So, after two weeks, about 3125 people were infected.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. In simple terms, if an exponential function tells us what the output is for a given power, a logarithmic function tells us what the power needs to be to get a certain output. They are written in the form of \( y = \log_b(x) \) where \(b\) is the base and \(x\) is the value we want to find the logarithm of. An essential characteristic of logarithms is that \( \log_b(b^x) = x \) and \( b^{\log_b(x)} = x \) for any \(x > 0\) and \(b > 0\), where \(b\) is not equal to 1.

Exponential Functions

Exponential functions are mathematical expressions where a number, usually denoted as the base, is raised to a variable exponent. In general form, an exponential function is represented as \( y = b^x \), where \(b\) is the base and \(x\) is the exponent. Unlike polynomial functions where the variable is raised to a constant power, in exponential functions, the power is variable. This leads to a rapid increase or decrease in the value of the function as \(x\) changes, which is why these functions are often used to model growth or decay processes such as population growth, radioactive decay, or, in our exercise, the spread of a virus. An important property is that the derivative of \(b^x\) is proportional to itself, which is fundamental to continuous growth and decay models.

When \(b=e\), a special mathematical constant approximately equal to 2.71828, the function \( y = e^x \) is known as the natural exponential function and is denoted by \(exp(x)\), often simplifying differential and integral calculus computations.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\), where \(e\) is Euler's number, approximately equal to 2.71828. It is a special case of logarithmic functions but is so common in mathematics and the sciences that it's often referred to simply as 'the logarithm.' The natural logarithm of a number \(x\) answers the question: 'To what power must we raise \(e\) in order to obtain the number \(x\)?' So, \(e^{\ln(x)} = x\) and conversely, \(\ln(e^x) = x\).

The natural logarithm has several properties that make it very useful, especially with growth processes and calculating time. It is continuously differentiable, and its derivative is \(\frac{1}{x}\), which is particularly handy in calculus. In our exercise, the natural logarithm is used to model the time it takes for a virus to infect a certain number of people, a common application in epidemiology.