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.

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