Question

A model for the number \(N\) of people in a college community who have heard a certain rumor is $$ N=P\left(1-e^{-0.15 d}\right) $$ where \(P\) is the total population of the community and \(d\) is the number of days that have elapsed since the rumor began. In a community of 1000 students, how many students will have heard the rumor after 3 days?

Short answer

362 students will have heard the rumor after 3 days.

Step-by-step solution

Identify Given Values

Identify the given values from the problem statement. Here, the total population of the community, \(P\), is 1000, and the number of days that have elapsed since the rumor began, \(d\), is 3 days.

Substitute Values into the Formula

Substitute the given values \(P = 1000\) and \(d = 3\) into the formula for \(N\): \[N = 1000 \times \left(1 - e^{-0.15 \times 3}\right)\]

Exponent Calculation

Calculate the exponent part \( -0.15 \times 3\): \[-0.15 \times 3 = -0.45\]

Evaluate the Exponential

Calculate the exponential \(e^{-0.45}\): \[e^{-0.45} \approx 0.6376\]

Substitute Exponential Back into the Formula

Substitute the evaluated exponential value back into the formula for \(N\): \[N = 1000 \times (1 - 0.6376)\]

Simplify the Expression

Simplify the expression to find \(N\): \[1 - 0.6376 = 0.3624\] Then, multiply by 1000: \[N = 1000 \times 0.3624 = 362.4\]

Interpret the Result

Since the number of students must be an integer, round the result to the nearest whole number: \[N \approx 362\]

Key concepts

Exponential Growth

Exponential growth is a process where a quantity increases rapidly over time. In mathematical terms, it's when a value grows at a rate proportional to its current size.
For example, if you start with 10 and it doubles every day, your value would follow an exponential growth pattern.
In the given problem, the number of people hearing the rumor spreads quickly, which fits well with exponential growth.
This model helps to predict how fast information spreads within a community, highlighting the speed and efficiency of word-of-mouth communication.

Substitution

Substitution is a basic yet crucial mathematical technique where you replace variables with known values to simplify a problem.
In this exercise, we start with the formula for rumor spread: \(N = P(1-e^{-0.15d})\).
We know that \(P\), the total population, is 1000, and \(d\), the number of days, is 3.
We substitute these values into the formula to get: \(N = 1000 \times (1 - e^{-0.15 \times 3})\).
Substitution makes complex problems easier to solve step by step.
Remember, carefully inserting the correct values reduces the chances of errors.

Exponential Function

An exponential function is a type of mathematical function that includes an exponent in its expression.
It's one of the key components in understanding exponential growth.
In the problem, the term \(e^{-0.15d}\) represents an exponential function where \(e\) is the base of the natural logarithm, approximately equal to 2.718.
Exponential functions are known for their property of rapid change: small changes in the exponent can lead to large changes in the overall value.
Here, we see how changing the number of days \(d\) affects the spread of the rumor in the community.

Population Model

A population model is a framework for simulating the dynamics of a population over time.
In this case, the model predicts how information spreads in a community.
The given formula, \(N = P(1-e^{-0.15d})\), is a form of population model.
It gives insight into how fast a rumor spreads to the community's total population \(P\).
The model is based on real-world observations and is useful in fields like epidemiology and social sciences.
Understanding this model helps us grasp complex real-world situations by simplifying them into manageable equations.