Question

The number of liver transplants performed in a particular country in year x is approximated by f left parenthesis x right parenthesisf(x)equals=negative 245.4 plus 2327 ln x−245.4+2327lnx where x greater than or equals 5x≥5 and xequals=5 corresponds to the year 1995. ​(a) estimate the number of transplants in 20252025. ​(b) find the rate of change for the number of transplants in 20252025.

Answer 1

(a). The function is written as: f(x) = -245.4 + 2327lnx

First, let's find the number of years between 1995 and 2025.
Number of years = 2025 - 1995 = 30
Since x = 5 represents the base year 1995, that mean we have to take x = 35.

f(35) = -245.4 + 2327ln(35) = 3,500 transplants by 2025

(b). The rate of change is determined through differential calculus.

f'(x) = 2327(1/x) = 2327/x
f'(35) = 2327/35 = 66.5

Thus, the rate of change of transplants is 66.5 per year.