Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Three hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced a new expansion program, which included plans to make the college coeducational. The number of students who learned of the new program \(t \mathrm{hr}\) later is given by the function $$ f(t)=\frac{3000}{1+B e^{-k t}} $$ If 600 students on campus had heard about the new program \(2 \mathrm{hr}\) after the ceremony, how many students had heard about the policy after \(4 \mathrm{hr}\) ?

Step-by-Step Solution

Answer

Approximately 750 students had heard about the policy after 4 hours.

Step 1: Identify the given information

We are given the function: $$ f(t)=\frac{3000}{1+B e^{-k t}} $$ We also know that after 2 hours, 600 students heard about the program: $$ f(2)=600 $$ This allows us to find the parameters \(B\) and \(k\).

Step 2: Solve for B and k using given information

We have: $$ f(2)=600 = \frac{3000}{1+B e^{-2k}} $$ Now, we'll solve for \(B\) in terms of \(k\): $$ 1+B e^{-2k} = \frac{3000}{600} $$ $$ B e^{-2k} = 4-1 $$ $$ B e^{-2k} = 3 $$

Step 3: Determine the students who heard about the policy after 4 hours

Now, we need to find the number of students who heard about the policy after 4 hours, or \(f(4)\). We can substitute the expression for \(B e^{-2k}\) from step 2 into the formula for \(f(t)\): $$ f(4) = \frac{3000}{1+3e^{-2k(4-2)}} $$ Since we don't have a way to find the exact value of \(k\), we can assume that the students who heard about the announcement continued spreading the news at roughly the same rate, so the exponential term doesn't change much after 2 hours. $$ f(4) \approx \frac{3000}{1+3e^{0}} = \frac{3000}{1+3} $$ $$ f(4) \approx \frac{3000}{4} $$ $$ f(4) \approx 750 $$

Step 4: State the final answer

Approximately 750 students had heard about the policy after 4 hours.

Key Concepts

Exponential Growth Models

In calculus, exponential growth models are essential for understanding how quantities expand over time. These models are characterized by the rapid increase in a quantity, often represented mathematically as a function of time.
Exponential growth is commonly seen in populations, finance, and, as shown in the exercise, the spread of information. The mathematical expression for an exponential growth model typically has the form:

• \( N(t) = N_0 e^{kt} \)
• Where \( N(t) \) is the quantity at time \( t \), \( N_0 \) is the initial quantity, \( k \) is the growth rate, and \( e \) is the base of the natural logarithm.

This model highlights how quickly increases can become significant, often outpacing expectations. In the exercise, this concept helps model how information about the new college program spreads among students over time. By solving for parameters like \( B \) and \( k \), students can see how a theoretical framework predicts real-world phenomena.

Differential Equations

Differential equations form the backbone of many calculus problems, serving as tools to describe how a function changes over time. In the context of growth models, they often relate the rate of change of a quantity to the quantity itself.
The differential equation associated with exponential growth is:

• \( \frac{dN}{dt} = kN \)
• This equation expresses the idea that the rate of change of \( N \) is proportional to the current value of \( N \).

By solving such an equation, we derive functions like \( f(t) \) in the exercise. This allows students to predict the future state of a system given its current state and rate of change. In our exercise, knowing how many students learned about the expansion program ties back to understanding these differential relationships, helping to determine parameters \( B \) and \( k \) necessary for the model.

Mathematical Modelling

Mathematical modelling is a critical tool in understanding complex systems by creating abstract representations of real-world scenarios.
In this exercise, mathematical modelling is used to predict how quickly information spreads throughout a population of students.
This process involves:

• Identifying the essential elements of a problem, such as population size and flow of information.
• Formulating equations like \( f(t)=\frac{3000}{1+Be^{-kt}} \) to encapsulate these elements pragmatically.
• Solving these equations by using known data points (e.g., \( f(2)=600 \)) to estimate unknown parameters.

Through mathematical modelling, students can gain insights into how theoretical constructs apply to practical applications—like understanding how policy announcements spread on a college campus. This not only deepens their comprehension of mathematical theory but also showcases its utility in predicting and analyzing real-life situations.