The number of people in a community who became infected during an epidemic t wee
ID: 3035949 • Letter: T
Question
The number of people in a community who became infected during an epidemic t weeks after its outbreak is given by the function
f left parenthesis t right parenthesis equals StartFraction 25 comma 000 Over 1 plus a e Superscript negative kt EndFractionf(t)=25,0001+aekt,
where
25 comma 00025,000
people of the community are susceptible to the disease. Assume that
50005000
people were infected initially and
87658765
had been infected by the end of the fourth week. Complete parts a. and
b..
a. Find the number of people infected after 8 weeks.
b. After how many weeks will
9 comma 8909,890
people be infected?
Explanation / Answer
f(t) = 25000/( 1 + ae^-kt)
Now 5000 people were infected intially at t=0 weeks
So, 5000 = 25000/( 1+ a)
1+a = 5 ; a = 4
f(t) = 25000/( 1+ 4e^-kt)
f(t) = 8765 ; t = 4
So, 8765 = 25000/( 1 +4e^-4k)
1 + 4e^-4k = 2.85
4e^-4k = 1.85
k = 0.7711/4 = 0.193
f(t) = 25000/( 1 + 4e^-0.193t)
t = 8 weeks ; f(8) = 13484.358 = 13484 people
after t = 9 weeks ; f(9) = 25000/(1 + 4e^-0.193*9)
= 14671 people