Questions: The number of computers N(t) (in millions) infected by a computer virus can be approximated by N(t) = 2.8 / (1 + 13 e^(-0.66 t)), where t is the time in months after the virus was first detected. Part 1 of 4 (a) Determine the number of computers initially infected when the virus was first detected. Round to the nearest ten thousand. Approximately 200,000 computers were infected when the virus was first detected. Part: 1 / 4 Part 2 of 4 (b) How many computers were infected after 4 months? Round to the nearest ten thousand. Approximately computers were infected after 4 months.

The number of computers N(t) (in millions) infected by a computer virus can be approximated by N(t) = 2.8 / (1 + 13 e^(-0.66 t)), where t is the time in months after the virus was first detected.

Part 1 of 4
(a) Determine the number of computers initially infected when the virus was first detected. Round to the nearest ten thousand.

Approximately 200,000 computers were infected when the virus was first detected.

Part: 1 / 4

Part 2 of 4
(b) How many computers were infected after 4 months? Round to the nearest ten thousand.

Approximately computers were infected after 4 months.

Transcript text: The number of computers $N(t)$ (in millions) infected by a computer virus can be approximated by $N(t)=\frac{2.8}{1+13 e^{-0.66 t}}$, where $t$ is the time in months after the virus was first detected. Part 1 of 4 (a) Determine the number of computers initially infected when the virus was first detected. Round to the nearest ten thousand. Approximately 200,000 $\square$ computers were infected when the virus was first detected. Part: $1 / 4$ Part 2 of 4 (b) How many computers were infected after 4 months? Round to the nearest ten thousand. Approximately $\square$ computers were infected after 4 months.

Solution

Solution Steps

Solution Approach

To determine the number of computers initially infected when the virus was first detected, we need to evaluate the function $ N(t) $ at $ t = 0 $. This will give us the initial number of infected computers. For the second part, we evaluate the function at $ t = 4 $ to find the number of infected computers after 4 months. Both results should be rounded to the nearest ten thousand.

Step 1: Initial Infection Calculation

To find the number of computers initially infected when the virus was first detected, we evaluate the function $ N(t) $ at $ t = 0 $:

\[ N(0) = \frac{2.8}{1 + 13 e^{-0.66 \cdot 0}} = \frac{2.8}{1 + 13} = \frac{2.8}{14} = 0.2 \]

This means that approximately $ 0.2 $ million computers were initially infected. Converting this to actual numbers gives:

\[ 0.2 \text{ million} = 200,000 \]

Step 2: Infection After 4 Months Calculation

Next, we calculate the number of computers infected after 4 months by evaluating $ N(t) $ at $ t = 4 $:

\[ N(4) = \frac{2.8}{1 + 13 e^{-0.66 \cdot 4}} \approx \frac{2.8}{1 + 13 \cdot 0.0498} \approx \frac{2.8}{1 + 0.6474} \approx \frac{2.8}{1.6474} \approx 1.4525 \]

This means that approximately $ 1.4525 $ million computers were infected after 4 months. Converting this to actual numbers gives:

\[ 1.4525 \text{ million} \approx 1,452,500 \]