In a town whose population is 2900, a disease creates an epidemic. The number of people [tex]\(N\)[/tex] infected [tex]\(t\)[/tex] days after the disease has begun is given by the function [tex]\[N(t) = \frac{2900}{1 + 21.8 e^{-0.5 t}}\][/tex]. Complete parts a) through d) below.

b) How many are initially infected with the disease [tex]\((t=0)\)[/tex]?
[tex]\[\text{(Round to the nearest whole number as needed.)}\][/tex]

c) Find the number infected after 2 days, 5 days, 8 days, 12 days, and 16 days.

The number infected after 2 days is [tex]\(\square\)[/tex].
[tex]\[\text{(Round to the nearest whole number as needed.)}\][/tex]



Answer :

Sure, let's go through finding the number of infected individuals at various days step-by-step using the given function [tex]\( N(t) = \frac{2900}{1 + 21.8 e^{-0.5t}} \)[/tex].

### Given Data:
- Population of the town, [tex]\( P = 2900 \)[/tex]
- Infection function, [tex]\( N(t) = \frac{2900}{1 + 21.8 e^{-0.5t}} \)[/tex]

### Part b) Initial infection when [tex]\( t = 0 \)[/tex]:

To find the number of initially infected individuals, substitute [tex]\( t = 0 \)[/tex] into the infection function:

[tex]\[ N(0) = \frac{2900}{1 + 21.8 e^{-0.5 \cdot 0}} \][/tex]

Since [tex]\( e^0 = 1 \)[/tex]:

[tex]\[ N(0) = \frac{2900}{1 + 21.8 \cdot 1} = \frac{2900}{1 + 21.8} = \frac{2900}{22.8} \approx 127 \][/tex]

So, the number of initially infected individuals [tex]\( N(0) \)[/tex] is approximately 127.

### Part c) Number of infected individuals at specific days: [tex]\( t = 2 \)[/tex], [tex]\( t = 5 \)[/tex], [tex]\( t = 8 \)[/tex], [tex]\( t = 12 \)[/tex], [tex]\( t = 16 \)[/tex]:

#### For [tex]\( t = 2 \)[/tex]:

[tex]\[ N(2) = \frac{2900}{1 + 21.8 e^{-0.5 \cdot 2}} \][/tex]

Calculating the exponent:

[tex]\[ e^{-1} \approx 0.36788 \][/tex]

Thus,

[tex]\[ N(2) = \frac{2900}{1 + 21.8 \cdot 0.36788} \approx \frac{2900}{1 + 8.0178} \approx \frac{2900}{9.0178} \approx 322 \][/tex]

So, after 2 days, approximately 322 individuals are infected.

#### For [tex]\( t = 5 \)[/tex]:

[tex]\[ N(5) = \frac{2900}{1 + 21.8 e^{-0.5 \cdot 5}} \][/tex]

Calculating the exponent:

[tex]\[ e^{-2.5} \approx 0.08208 \][/tex]

Thus,

[tex]\[ N(5) = \frac{2900}{1 + 21.8 \cdot 0.08208} \approx \frac{2900}{1 + 1.788} \approx \frac{2900}{2.788} \approx 1040 \][/tex]

So, after 5 days, approximately 1040 individuals are infected.

#### For [tex]\( t = 8 \)[/tex]:

[tex]\[ N(8) = \frac{2900}{1 + 21.8 e^{-0.5 \cdot 8}} \][/tex]

Calculating the exponent:

[tex]\[ e^{-4} \approx 0.01832 \][/tex]

Thus,

[tex]\[ N(8) = \frac{2900}{1 + 21.8 \cdot 0.01832} \approx \frac{2900}{1 + 0.399376} \approx \frac{2900}{1.399376} \approx 2072 \][/tex]

So, after 8 days, approximately 2072 individuals are infected.

#### For [tex]\( t = 12 \)[/tex]:

[tex]\[ N(12) = \frac{2900}{1 + 21.8 e^{-0.5 \cdot 12}} \][/tex]

Calculating the exponent:

[tex]\[ e^{-6} \approx 0.00248 \][/tex]

Thus,

[tex]\[ N(12) = \frac{2900}{1 + 21.8 \cdot 0.00248} \approx \frac{2900}{1 + 0.054064} \approx \frac{2900}{1.054064} \approx 2751 \][/tex]

So, after 12 days, approximately 2751 individuals are infected.

#### For [tex]\( t = 16 \)[/tex]:

[tex]\[ N(16) = \frac{2900}{1 + 21.8 e^{-0.5 \cdot 16}} \][/tex]

Calculating the exponent:

[tex]\[ e^{-8} \approx 0.00034 \][/tex]

Thus,

[tex]\[ N(16) = \frac{2900}{1 + 21.8 \cdot 0.00034} \approx \frac{2900}{1 + 0.007412} \approx \frac{2900}{1.007412} \approx 2879 \][/tex]

So, after 16 days, approximately 2879 individuals are infected.

### In Summary:

- The number infected after 2 days is approximately 322.
- The number infected after 5 days is approximately 1040.
- The number infected after 8 days is approximately 2072.
- The number infected after 12 days is approximately 2751.
- The number infected after 16 days is approximately 2879.