So a computer virus has been created via Corporate e-mail. Let t=0 be the time of the first 100 infections, and at t=17 minutes the population of infected computers grows to 200. Assume the anti-virus companies arent able to identify the virus or slow its progress for 24 hours, allowing the virus to grow exponentially.
1. What will the population of the infected computer be after 1 hour?
(So i have to set up the equation for P(t)=P_0e^kt, where t=0 for the first 100 infections and t=17 when it grows to 200...but i don't know where to start or set it up) Help please.
To set up the equation for the population of infected computers, we can take the initial population at t = 0 as P₀ = 100, and at t = 17 minutes, it grows to 200. We need to find the value of k to determine the growth rate.
Using the formula P(t) = P₀ * e^(k*t), we can substitute the given values to obtain the following equation:
200 = 100 * e^(k * 17)
To solve for k, we need to isolate it on one side of the equation. Divide both sides of the equation by 100 to get:
2 = e^(k * 17)
Now, to find the value of k, take the natural logarithm of both sides of the equation:
ln(2) = k * 17
Now divide ln(2) by 17 to get the value of k:
k ≈ ln(2) / 17
Once we know the value of k, we can substitute it into the equation P(t) = P₀ * e^(k*t) to determine the population of infected computers after 1 hour. Since we have one hour (60 minutes), substitute t = 60 into the equation:
P(60) = 100 * e^((ln(2)/17) * 60)
Calculating the expression on the right-hand side will give you the final answer:
P(60) ≈ 100 * e^(60 * ln(2) / 17)
To set up the equation for the population of infected computers, we can use the exponential growth formula:
P(t) = P0 * e^(kt),
where:
- P(t) represents the population of infected computers at time t,
- P0 is the initial population of infected computers at t = 0,
- e is Euler's number (approximately 2.71828),
- k is the growth rate (which we need to determine).
From the given information, we know that at t = 0, there were 100 infections (P0 = 100), and at t = 17 minutes, the population grew to 200. We can substitute these values into the equation to solve for k:
200 = 100 * e^(k * 17).
To isolate the exponential term, divide both sides by 100:
2 = e^(k * 17).
Now, we can take the natural logarithm (ln) of both sides:
ln(2) = k * 17.
Finally, solve for k by dividing both sides by 17:
k = ln(2) / 17 ≈ 0.040598.
Now that we have determined k, we can use it to find the population of infected computers after 1 hour (60 minutes). Substitute t = 60 into the equation:
P(60) = 100 * e^(0.040598 * 60).
Evaluate the expression on the right-hand side to find the population of infected computers after 1 hour.
Note: It's important to keep in mind that this calculation assumes uninterrupted exponential growth, which may not be accurate in real-world scenarios.