Question

the function ( n(t)=\frac{20,000}{1 + 30e^{-2.0t}} ) describes the number of people, ( n(t) ), who become ill with a virus ( t ) weeks after its initial outbreak in a town with 20,000 inhabitants. the horizontal asymptote in the graph indicates that there is a limit to the epidemics growth. complete parts (a) through (c) below. a. how many people became ill with the virus when the epidemic began? (when the epidemic began, ( t = 0 ).) when the epidemic began, approximately ( square ) people were ill with the virus. (round to the nearest person as needed.)

Explanation:

Step1: Substitute t = 0 into the function

We have the function \( N(t)=\frac{20000}{1 + 30e^{-2.0t}} \). When \( t = 0 \), we substitute \( t \) with 0 in the function. So we get \( N(0)=\frac{20000}{1+30e^{-2.0\times0}} \).

Step2: Simplify the exponent and the denominator

First, calculate the exponent: \( - 2.0\times0 = 0 \). Then, \( e^{0}=1 \). So the denominator becomes \( 1 + 30\times1=1 + 30 = 31 \). Now the function is \( N(0)=\frac{20000}{31} \).

Step3: Calculate the value

Calculate \( \frac{20000}{31}\approx645.16 \), and rounding to the nearest person, we get approximately 645.

Answer:

645