# A computer virus is spreading at an exponential rate in a network of 1, 000 computers. Initially,...

## Question:

A computer virus is spreading at an exponential rate in a network of 1, 000 computers. Initially, 2 computers are infected. After 1 day has passed, 5 computers are infected. To the nearest number of whole days, how many days will it take until 25% of the computers are infected?

## Exponential Growth:

As we have been told that the virus is spreading at an exponential rate, we will be using an exponential model. This model will be:

{eq}C(t) =C(o) e^{kt} {/eq}

As usual, the first step is to find the rate of growth k. This is shown below.

The computer virus is spreading at an exponential rate. This means we can use the below function to estimate the number of computers infected after t days.

{eq}C(t) =C(o) e^{kt} {/eq}

Here, C(t) is the number of computers infected after t days when initially C(o) computers are infected. k is the rate at which the computers are getting infected. We already know that initially 2 computers are infected. The value of k can be found as follows using the information provided.

{eq}C(t) =2e^{kt} \\ \text {When t=1, C=5}\\ \Rightarrow 2e^{k} =5\\ k=\ln2.5 {/eq}

The model now is:

{eq}C(t) =2e^{t\ln2. 5t}\\ {/eq}

25% of the total computers means 250. Therefore, we want to find the time after which 250 computers will be infected. This can be found as follows.

{eq}C(t) =2e^{t\ln2. 5}=250\\ t\ln2.5=\ln125\\ t\approx 5\,\text{days} {/eq}