Homework SourseThe population of a region is growing exponentially There we
The population of a region is growing exponentially. There were 40 million people in 1980 (when t=0) and 75 million people in 1990. Find an exponential model for the population (in millions of people) at any time t, in years after 1980.
P(t)=
Predicted population in the year 2000 = million people.
P(t)=
Predicted population in the year 2000 = million people.
Solution
The general exponential function is P(t) = a b^t. So you know these facts (numbers are millions): 40 = a b^0 , and since b^0 = 1, you know a = 40 72 = 40 b^10 so to solve for b, divide by 40 then do the tenth root of each side (72/40) ^ 0.1 = b. 72/40 reduces to 1.8. So the equation is P(t) = 40 • 1.8 ^ 0.1 t 2000 is 20 years after 1980 so in 20 years you\'d predict P(20) = 40 • 1.8 ^( 0.1•20) so see what that is. To double, P(t) would be 80 so 80 = 40 • 1.8 ^ 0.1t Divide by 40 2 = 1.8 ^ 0.1 t log both sides log 2 = 0.1t log 1.8 so t = log 2 ÷ (0.1 log 1.8) so see what that is. I got 11.8 years
