Use these data and the Law of Uninhibited Growth to write a formula for the world population P = P ( t) .

Question

The world population on July 1, 1959, was approximately 2.983435×109 persons and had a doubling time of D = 40 years. Use these data and the Law of Uninhibited Growth to write a formula for the world population P = P ( t) .

Answer 1

not that it is a good model but

want f(t)=Ab^t,,,is general exponential function (the constant e is not needed)

know A=3x10⁹,,,,when t=0 b⁰=1

know 6x10⁹=3x10⁹b⁴⁰,,,,,t=40 f(t) is 2x3x10⁹

b=2^1/40 ,,,,dividing and taking 40th root

f(t)=(3x10⁹)(2^(1/40))^t

f(t)=(3x10⁹)2^t/40,,,,,,,,property of exponents,,,,exponent to exponent is multiplication of the values

P(t)=(3x10⁹)2^(t/40),,,,t measured in years t=0 7/1/1959

in 2039, in this model P(t)=12x10⁹

a better model at this point might be linear growth (unchecked)

P(t)=(8x10⁹)+((1/12)x10⁹)t,,,,,t measured in years t=0 11/15/2022

in 2040, in this model P(t)=9.5x10⁹.

Even going back to 1959, linear model P(t)=(3x10⁹)+((1/12)x10⁹)t works kind of well.

It would say in 2019 P(t)=8x10⁹ which is kind of close.

Answer 2

2 = e^40k

ln 2 = 40k

Solve for k

P(t) = (2.983435×10⁹)(2)^kt, substitute k from above.