algebra 2 problem

Question

The recent Ebola outbreak in Africa began in early 2014. According to the CDC by April 2014, 122 cases had been reported; and by June 2014, 344 cases had been reported. Assuming the virus spreads expeditiously, predict the number of cases and June 2015

Answer 1

I assume the question actually says to assume the virus spreads exponentially, rather than expeditiously.  Simple continuous exponential growth can be modeled by P(t) = C * e^rt, with C and r being constants.  If you know the size P at two times, you can determine r by dividing the sides:

 

P₂/P₁ = (C*e^rt₂)/(C*e^rt₁)

P₂/P₁ = e^rt₂/e^rt₁

P₂/P₁ = e^r(t₂-t₁)

ln (P₂/P₁) = r(t₂-t₁)

r = ln (P₂/P₁)/(t₂-t₁)

 

P₁ is 122 cases; we don't have a strict zero, so we can just call April 2014 our time 0.

P₂ is 344 cases, two months later, so our t₂ is 2 months.

 

Thus, r = ln(344/122) / 2 = ~.518

and C can be determined by 122 = C*e^r(0):  C is 122.

 

June 2015 is 14 months after our time zero, so

P = 122*e^(.518*14) = 172,133 cases.