# 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.

## Answer and Explanation:

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}