Question

He population of a region is growing exponentially. there were 40 million people in 1980 (when t=0) and 55 million people in 1990. find an exponential model for the population (in millions of people) at any time t, in years after 1980.

Answer 1

P(t) = 40(2)^(kt) 
when t=10, (1990), N = 55 
55 = 40(2)^(10k) 
1.25 = 2^(10k) 
take the ln of both sides, hope you remember your log rules 
10k = ln 1.25/ln 2 
10k = .32193 
k = .032193 

so P(t) = 40(2)^(.032193t) 

in 2000, t = 20 
P(20) = 40(2)^(.032193(20)) 
= 62.5 million 

for the formula 
P(t) = a(2)^(t/d), d = the doubling time 
so changing .032193t to t/d 
= .032193t 
= t/31.06 

so the doubling time is 31.06 

another way would be to set 
80 = 40(2)^(.032193t) 
2 = (2)^(.032193t) 
.032193t = ln 2/ln 2 = 1 
t = 31.06