Problem 48 - College Algebra

Question

It is 8: 00 P.M. and West Side Story is scheduled to begin. When the curtain does not go up, a rumor begins to spread through the 400-member audience: The lead roles of Tony and Maria might be understudied by Anthony Hopkins and Jodie Foster. The function $$f(x)=\frac{400}{1+399(0.67)^{x}}$$ models the number of people in the audience, \(f(x),\) who have heard the rumor \(x\) minutes after \(8: 00 .\) Use this function to solve. Evaluate \(f(20)\) and describe what this means in practical terms.

Step-by-Step Solution

Answer

By calculating \( f(20) \) using the provided function, the estimated number of people in the audience who have heard the rumour 20 minutes after it started will be obtained. The exact number will be determined by the calculation.

Step 1: Understanding the function

The function \( f(x) = \frac{400}{1 + 399(0.67)^x} \) models the number of people in the crowd who have heard the rumour \( x \) minutes after 8:00PM. Here, \( f(x) \) is the number of people who have heard the rumour and \( x \) is the time in minutes since 8:00PM.

Step 2: Substituting the given value into the function

We are asked to find \( f(20) \), this means we substitute \( x \) with 20 in the function: \( f(20) = \frac{400}{1 + 399(0.67)^{20}} \).

Step 3: Evaluating the function

Computing the exponential part first, evaluate \( (0.67)^{20} \). Then multiply the result with 399 and add 1. Finally, divide 400 with this number to compute the value of \( f(20) \).

Step 4: Interpreting the result

The result \( f(20) \) represents the number of people in the audience who have heard the rumour 20 minutes after 8:00 PM. It's crucial to remember that the model is an approximation, so the result should likewise be interpreted as an estimate, not an exact number of people.

Key Concepts

College Algebra

College algebra is considered a foundation stone in the world of mathematics, as it introduces students to a variety of abstract concepts and mathematical formats that have practical applications, such as exponential functions.

In the context of our exercise, college algebra deals with functions, their structures, and how to work with them, setting the groundwork for understanding more complex mathematical models. It enables students to tackle real-world problems, like the spread of a rumor among a crowd, by using algebraic formulas and function evaluation.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-life scenarios. In our example, the function

\( f(x) = \frac{400}{1 + 399(0.67)^x} \)

models the dissemination of a rumor in an audience over time. This representation transforms the abstract concept of rumor spread into a calculable form, allowing us to predict how many people will have heard the rumor at any given time after the curtain was supposed to go up. Mathematical models rely on assumptions, such as the initial number of people (in this case, 400) and the rate at which the rumor spreads, which in this function is controlled by the base of the exponential term, 0.67.

Exponential Functions

Exponential functions, which take the form \( y = ab^x \), where \( a \) and \( b \) are constants, and \( x \) is the variable, are indispensable in describing situations where change occurs at a rate proportional to the current value, such as population growth, radioactive decay, or, as in our exercise, rumor spreading.

The function

\( f(x) = \frac{400}{1 + 399(0.67)^x} \)

shows an initial rapid spread that slows over time, characteristic of an exponential decay model. Recognizing the type of exponential function helps students anticipate the function's behavior without needing to plot it, which is a powerful analytical skill in both mathematics and real-life problem-solving.

Function Evaluation

Function evaluation is a basic yet crucial concept in algebra that involves substituting a number into the function's variable and computing its value. Knowing how to accurately evaluate functions allows students to find specific solutions to problems modeled by those functions.

For instance, evaluating \( f(20) \) for our function involved substituting 20 for \( x \), calculating the result, and interpreting it as the estimated number of people who had heard the rumor 20 minutes after 8:00 PM. Instruction on the subject includes understanding orders of operations and the influence of different function components on the outcome.