Problem 43 - Intermediate Algebra

Question

Computer Viruses. Suppose the number of computers infected by the spread of a virus through an e-mail is described by the exponential function \(c(t)=5(1.034)^{t},\) where \(t\) is the number of minutes since the first infected e-mail was opened. a. Graph the function. Scale the \(t\) -axis from 0 to \(400,\) in units of \(50 .\) Scale the \(c(t)\) -axis from 0 to \(800,000\) in units of \(100,000\) b. Use the function to determine the number of infected computers in 8 hours, which is 480 minutes.

Step-by-Step Solution

Answer

After 8 hours (480 minutes), approximately 771,677 computers are infected.

Step 1: Understanding the Function

The function provided is \(c(t) = 5(1.034)^{t}\). This exponential function describes the number of computers infected over time, where \(t\) is the time in minutes since an infected email was opened.

Step 2: Determine the Rate of Infection

The base of the exponential function, \(1.034\), indicates that each minute, the number of infected computers increases by 3.4% over the previously infected count.

Step 3: Graphing the Function

Plot the graph with the \(t\)-axis ranging from 0 to 400 (in increments of 50) and the \(c(t)\)-axis from 0 to 800,000 (in increments of 100,000). Because the function is exponential, expect a sharp increase on the graph, especially as \(t\) increases.

Step 4: Calculate Infected Computers at 480 Minutes

Substitute \(t = 480\) into the function: \[c(480) = 5(1.034)^{480}\]This calculation will provide the number of computers infected after 8 hours (480 minutes).

Step 5: Solve the Exponential Expression

Using a calculator, compute as follows:1. Compute \((1.034)^{480}\).2. Multiply the result by 5 to find \(c(480)\).

Step 6: Interpret the Result

The result of the calculation in Step 5 provides the number of computers infected after 480 minutes. This value reflects the rapid spread of the virus by email.

Key Concepts

Exponential Functions

Exponential functions are a fundamental concept in mathematics and are vital in modeling scenarios where quantities grow or shrink at a constant relative rate. These functions can be represented in the form \(f(t) = a(b)^t\), where \(a\) is the initial quantity and \(b\) is the growth (or decay) factor. In the context of computer virus infections, an exponential function describes how quickly a virus can spread by doubling over regular intervals.
The base \(b\) in an exponential function signals the rate of growth or change. If \(b > 1\), the function models exponential growth. Here in our function, \(b = 1.034\), implying that for every minute past the initial infection, the number of infected computers grows by 3.4%.
When using exponential functions to model real-world scenarios like virus infections, it’s critical to understand:

• Initial quantity or starting point (\(a\)): Determines how many units are initially affected.
• Growth or decay rate (\(b\)): Indicates how quickly the change happens. In our case, it’s a growth as \(b > 1\).
• Time factor (\(t\)): Represents the passage of time affecting the change in quantity.

Understanding these components helps predict future outcomes based on current trends, which in this problem, predicts virus spread over time.

Graphing Exponential Functions

Graphing an exponential function involves plotting the curve based on its equation and understanding its general shape. Exponential graphs are typically characterized by their rapid rise or fall.
For the problem at hand with the function \(c(t) = 5(1.034)^t\), the graph will exhibit exponential growth. You can expect:

• A slow start at the beginning, due to the initial quantity being small.
• An accelerated slope as \(t\) increases, depicting a rapid rise in the number of infected computers.
• The graph never touches the horizontal axis (the \(t\)-axis), indicating that zero infections are never realistically reached unless \(t\) is negative or the growth factor is less than one.

When plotting:

• Set the \(t\)-axis: Here, from 0 to 400 minutes based on the problem requirements. This helps visualize the short-term and medium-term progression.
• Set the \(c(t)\)-axis: From 0 to 800,000 to accommodate the anticipated rise in infections.

These choices allow for a full display of the function's growth, demonstrating how a small-scale infection can quickly escalate given the conditions modeled by the exponential function. It's crucial to accurately select your scales to avoid misinterpretation of the graph's growth nature.

Infection Modeling

Infection modeling using exponential functions plays a significant role in understanding how diseases or digitally modeled infections (like computer viruses) spread over time. Such modeling uses mathematical concepts to predict possible future conditions based on current data.
The problem illustrates infection modeling by factoring:

• Initial Conditions: The current status of infections, represented by the value at \(t = 0\), or when the first computer gets infected.
• Transmission Rate: The growth factor \(1.034\), which in human health modeling could translate to contact or reproduction rates.
• Time Frame: You can see dramatic changes, similar to biological infections, over short periods, indicated by how quickly the number of infected computers rises to substantial levels, such as what is calculated at \(t = 480\) minutes.

Effective infection modeling helps authorities and organizations decide on interventions and measures necessary to contain or mitigate the spread of infections. It enables planning for possible future scenarios and allocating resources to minimize impact. In digital contexts, like the spread of a computer virus, comprehensive modeling can inform cybersecurity measures and the urgency of patching or protecting systems.