The number of people in a community who became infected duri

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?

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