There was one heart transplant in 1967 by Dr. Christian Bernard of South Africa. In 1987, there were 1,418 heart transplants

Question

a.) Find the exponential growth function that fits this data

b.) Use this function to predict the number of heart transplants in 2015

Answer 1

Casandra,

 

Since the problem states an exponential growth function, let's use 

 

N(T) = C e^kT , where

  

        T = years since 1967, so T=0 in 1967

   N(T) = number of heart transplants

        C = a constant applied to the exponential

        K = a different constant that modifies T before computing the exponential

 

Find the constant from the initial conditions.

In 1967, T=0 and N(0) = 1 , so  1 = C e⁽⁰⁾ = C since e⁽⁰⁾ = (2.718)⁽⁰⁾ = 1

In 1987, T =20 and N(20) = 1418 = e^(20k) ;  take logarithms of both sides to solve for k

ln (1418) = (20k) ln e = 20k (1) = 20k , so k= (1/20) ln(1418) = (1/20) (7.257) = 0.36285   (1/years)

 

(a) The growth formula then is   N(T) = e ^{[(0.36285) T]} .

 

(b) In 2015,  T = 2015 - 1967 = 48 years

         So  N(48) = e ^{[(0.36285)(48)]}    =  e ^[17.4168] = 36,645,456  heart transplants in 2015.

 

Ridiculous! You couldn't find that many working hearts to replace diseased ones.  So an exponential formula assumption for growth is not realistic for long time periods.  It might be okay for early growth, but then another formula would have to take over.