Problem 55 - Algebra 2

Question

The number of people \(N\) who will receive a forwarded e-mail can be approximated by \(N=\frac{P}{1+(P-S) e^{-0.35 t}},\) where \(P\) is the total number of people online, \(S\) is the number of people who start the e-mail, and \(t\) is the time in minutes. Suppose four people want to send an e-mail to all those who are online at that time. If there are 156,000 people online, how many people will have received the e-mail after 25 minutes?

Step-by-Step Solution

Answer

Approximately 6,059 people will have received the e-mail after 25 minutes.

Step 1: Identify the given values

From the problem, we know that the total number of people online is given by \(P = 156,000\). The number of people who started forwarding the e-mail is \(S = 4\), and we want to find \(N\) after \(t = 25\) minutes.

Step 2: Substitute the values into the formula

We substitute \(P = 156,000\), \(S = 4\), and \(t = 25\) into the formula: \[N = \frac{156,000}{1 + (156,000 - 4) e^{-0.35 \times 25}}\].

Step 3: Calculate the exponent and the expression inside the denominator

Calculate \(-0.35 \times 25 = -8.75\). Then calculate the expression inside the denominator:\[(156,000 - 4)e^{-8.75} = 155,996 imes e^{-8.75}\].

Step 4: Compute the exponential expression

Calculate \(e^{-8.75}\) using a calculator or computational tool to get approximately \(0.00015865\). Then multiply by \(155,996\):\[155,996 \times 0.00015865 \approx 24.749\].

Step 5: Calculate the entire denominator

Substitute the result into the denominator expression:\[1 + 24.749 = 25.749\].

Step 6: Find the final number of people \(N\)

Substitute back into the formula to find \(N\):\[N = \frac{156,000}{25.749} \approx 6,059 \].

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. This allows them to model phenomena that grow or decay at accelerating rates, such as populations, radioactive decay, and viral spreads. These functions typically take the form of \(f(t) = a \cdot b^t\), where \(a\) is a starting value and \(b\) signifies the growth or decay factor.
In the context of our e-mail forwarding problem, the formula \(N=\frac{P}{1+(P-S) e^{-0.35 t}}\) represents an exponential function. Here, \(e^{-0.35t}\) showcases how the forwarding process slows down over time, as fewer new recipients receive the e-mail. The function uses mathematical constants and operations (specifically, the base \(e\)) to calculate how many people eventually receive the e-mail.
This kind of mathematical function is crucial in understanding how quickly and broadly information can spread within a network.
In practice, these models can help predict not just viral emails but also the spread of trends or infections through populations, highlighting the significance of exponential behavior in real-world scenarios.

Mathematical Modeling

Mathematical modeling is the process of creating a mathematical representation of a real-world scenario to predict and analyze its behavior. This approach is used in various fields, from economics to biology, helping experts to make informed decisions.
In the e-mail scenario, the given formula acts as a mathematical model to predict how many people will receive the forwarded message over time. This model takes into account several factors: the total online population \(P\), the initial senders \(S\), and the time \(t\). Each element interacts within the formula to describe and predict the spread dynamics of the email.
The model's accuracy in predicting real-world outcomes depends on how well it reflects the actual scenario processes. For instance, the constant \(-0.35\) in the formula represents the rate at which the forwarding process slows down, which is an estimated factor based on assumed conditions.
Creating effective models involves understanding the real-world phenomenon deeply, ensuring that all significant influencing factors are incorporated into the mathematical representation.

Computational Tools

Computational tools are programs and software applications used to perform complex calculations quickly and accurately, aiding mathematicians and scientists with models involving exponential functions. These tools range from scientific calculators to advanced software like MATLAB or Python libraries.
For the exponential calculation of \(e^{-8.75}\) in the problem, a computational tool is invaluable. Computing this manually with high precision would be complex and prone to error, while a calculator or software can do it rapidly.

• Scientific calculators: provide a straightforward way to perform exponential and logarithmic calculations.
• Software programs: such as MATLAB or Python's NumPy, offer more robust solutions for handling large sets of calculations and visualizing data.

Using computational tools makes it easier to test complex models or scenarios by automating the calculation process and reducing human error. In the e-mail exercise, these tools allow for quick adjustments and re-calculations if parameters change, providing flexibility and efficiency in mathematical modeling.