Certainly! Let's solve the problem step-by-step, using the function provided to determine the number of people initially infected when [tex]\( t = 0 \)[/tex].
The function given is:
[tex]\[ N(t) = \frac{3400}{1 + 24.8 e^{-0.7 t}} \][/tex]
### Step-by-Step Solution:
1. Identify the Initial Condition:
We need to find the initial number of infected people, which means evaluating [tex]\( N(t) \)[/tex] at [tex]\( t = 0 \)[/tex].
2. Substitute [tex]\( t = 0 \)[/tex] Into the Function:
Plugging [tex]\( t = 0 \)[/tex] into the function, we get:
[tex]\[
N(0) = \frac{3400}{1 + 24.8 e^{-0.7 \cdot 0}}
\][/tex]
3. Simplify the Exponential Term:
Since any number raised to the power of zero is 1:
[tex]\[
e^{0} = 1
\][/tex]
4. Simplify the Denominator:
Substituting [tex]\( e^{0} = 1 \)[/tex] into the denominator, we get:
[tex]\[
1 + 24.8 \cdot 1 = 1 + 24.8 = 25.8
\][/tex]
5. Calculate the Initial Number of Infected People [tex]\( N(0) \)[/tex]:
Now, we simply divide the population by this number:
[tex]\[
N(0) = \frac{3400}{25.8}
\][/tex]
6. Perform the Division:
Divide 3400 by 25.8 to find the initial number of infected people:
[tex]\[
N(0) = 131.78
\][/tex]
7. Round to the Nearest Whole Number:
Since we are asked to round to the nearest whole number:
[tex]\[
N(0) \approx 132
\][/tex]
### Conclusion:
So, the number of people initially infected with the disease when [tex]\( t = 0 \)[/tex] is [tex]\( \boxed{132} \)[/tex].
