Question

the function $n(t) = \frac{14,000}{1 + 999e^{-t}}$ models the number of people in a small town who have caught the flu $t$ weeks after the initial outbreak. step 2 of 2 : how many people have caught the flu after 10 weeks? round to the nearest person.

Explanation:

Step1: Substitute \( t = 10 \) into the function

We have the function \( N(t)=\frac{14000}{1 + 999e^{-t}} \). Substitute \( t = 10 \) into it, so we get \( N(10)=\frac{14000}{1 + 999e^{-10}} \).

Step2: Calculate the value of \( e^{-10} \)

We know that \( e^{-10}\approx4.539993\times 10^{-5} \) (using the property of the exponential function \( e^x=\sum_{n = 0}^{\infty}\frac{x^n}{n!} \), for \( x=- 10 \), we can also use a calculator to get this approximation).

Step3: Calculate the denominator

First, calculate \( 999e^{-10}\approx999\times4.539993\times 10^{-5}\approx0.0453545 \). Then the denominator \( 1 + 999e^{-10}\approx1+0.0453545 = 1.0453545 \).

Step4: Calculate \( N(10) \)

Now, \( N(10)=\frac{14000}{1.0453545}\approx13392.5 \). Rounding to the nearest person, we get \( N(10)\approx13393 \).

Answer:

\( 13393 \)