Posted by **Z32** on Tuesday, October 6, 2009 at 11:58pm.

The population of a region is growing exponentially. There were 40 million people in 1980 (when t=0) and 50 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.

What is the doubling time?

Doubling time = _______ years.

- Calculus -
**Reiny**, Wednesday, October 7, 2009 at 12:22am
Since one of the questions deals with "doubling time", I will use 2 as the base.

(Of course we could use any base, many mathematicians would automatically use e as the base.)

P(t) = a(2)^(kt) , where P(t) is the population in millions, a is the beginning population, and t is the time in years.

clearly , a = 40

P(t) = 40(2)^(kt)

when t=10, (1990), N = 50

50 = 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

- Calculus -
**bobpursley**, Wednesday, October 7, 2009 at 12:05pm
Curiously, this is very close to the current population doubling time for India.

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