MODERATE EXERCISES-Logarithms and Modelling

(1) In 1967, Dr. Christian Barnard, of South Africa, staggered the world by performing the first heart transplant. There was 1 transplant in 1967. In 1987, there were 1418 such transplants.

1. Assume that the number of heart transplants performed per year grow exponentially. Using the formula N = N₀ e^kt
   find the exponential growth function that fits the data.
2. Use your answer in part (a) to predict the number of heart transplants in 1996.

SOLUTION:

1. If t=0 in 1967, then N₀=1. Then when t=20, N=1418. Substituting into the formula (function) for N, we get: 1418 = 1 e^k(20)
   Taking ln of both sides: ln(1418) = ln[e^20k]
   So then ln(1418) = 20k or k = ln(1418) / 20. That is, k = 0.3629 (to 4 decimal places).

2. We know N₀=1 and now we know k, so the formula for N is: N = e^0.3629t
   Evaluating this when N=29, we get: N = e^0.3629(29) = 37,201
   The model predicts approximately 37,201 heart transplants in 1996.

   
   

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