A Network Administrator Uses The Function $f(x) = 5^x$ To Model The Number Of Computers A Virus Spreads To After $x$ Hours. If There Are 1,000 Computers On The Network, About How Many Hours Will It Take For The Virus To Spread To The

**A Network Administrator's Dilemma: Modeling the Spread of a Virus**

Understanding the Function

The function $f(x) = 5^x$ is used to model the number of computers a virus spreads to after $x$ hours. This function is an exponential function, which means that the number of computers infected grows rapidly as the number of hours increases.

The Initial Condition

There are 1,000 computers on the network, and we want to know how many hours it will take for the virus to spread to all of them. This is a classic problem of exponential growth, and we can use the function $f(x) = 5^x$ to model it.

The Question

How many hours will it take for the virus to spread to all 1,000 computers?

The Solution

To solve this problem, we need to find the value of $x$ that makes $f(x) = 1,000$. In other words, we need to solve the equation $5^x = 1,000$.

Step 1: Take the Logarithm

To solve the equation $5^x = 1,000$, we can take the logarithm of both sides. This will allow us to use the properties of logarithms to simplify the equation.

$$\log(5^x) = \log(1,000)$$

Step 2: Use the Power Rule

Using the power rule of logarithms, we can rewrite the left-hand side of the equation as:

$$x \log(5) = \log(1,000)$$

Step 3: Solve for x

Now we can solve for $x$ by dividing both sides of the equation by $\log(5)$:

$$x = \frac{\log(1,000)}{\log(5)}$$

Step 4: Calculate the Value

Using a calculator, we can calculate the value of $x$:

$$x \approx \frac{3.0000}{0.69897} \approx 4.30$$

The Answer

Therefore, it will take approximately 4.30 hours for the virus to spread to all 1,000 computers.

Q&A

Q: What is the purpose of using the function $f(x) = 5^x$ to model the spread of a virus?

A: The function $f(x) = 5^x$ is used to model the number of computers a virus spreads to after $x$ hours. This function is an exponential function, which means that the number of computers infected grows rapidly as the number of hours increases.

Q: How many computers are on the network?

A: There are 1,000 computers on the network.

Q: How many hours will it take for the virus to spread to all 1,000 computers?

A: It will take approximately 4.30 hours for the virus to spread to all 1,000 computers.

Q: What is the significance of the logarithm in solving the equation $5^x = 1,000$?

A: The logarithm is used to simplify the equation and make it easier to solve. It allows us to use the properties of logarithms to rewrite the equation in a more manageable form.

Q: How can the function $f(x) = 5^x$ be used in real-world applications?

A: The function $f(x) = 5^x$ can be used to model a wide range of real-world phenomena, including population growth, financial growth, and the spread of diseases.

Q: What are some potential limitations of using the function $f(x) = 5^x$ to model the spread of a virus?

A: Some potential limitations of using the function $f(x) = 5^x$ to model the spread of a virus include the assumption of exponential growth, the lack of consideration for other factors that may affect the spread of the virus, and the potential for overestimation or underestimation of the number of computers infected.

Q: How can the function $f(x) = 5^x$ be modified to better model the spread of a virus?

A: The function $f(x) = 5^x$ can be modified to better model the spread of a virus by incorporating additional factors that may affect the spread of the virus, such as the number of computers that are already infected, the rate at which new computers are infected, and the rate at which computers are removed from the network.