worlds population

Question

The worlds population in 1950 and 2013

 

Yrs.            Population

since.         (billions)

1950

 

  0                  2.555

63.                    7.2

 

Determine the perimeter (A,k) in exponential function (P(t)= Ae^kt) that fits world population (P(t)) to time(t)

 

 

Using this function calculate the current tripling time, for how long it takes for current population to triple?

Answer 1

We can find A and k by plugging in values. Start with the data from 1950, plugging 0 in for t and 2.555 in for P(t):

 

2.555 = Ae^k(0)

2.555 = Ae⁰

2.555 = A(1)

2.555 = A

 

Now that we know A, we can plug in the data from 2013, t = 63 and P(t) = 7.2, to find k:

 

7.2 = 2.555e^k(63)

2.818 = e^63k

ln(2.818) = 63k

1.036 = 63k

0.01644 = k    (approximately)

 

The final equation is P(t) = 2.555e^0.01644t

 

To find out how long it will take for the current population to triple, we need to know what x is when 2.555e^0.01644(t+x) = 3(2.555e^0.01644t). Let's solve:

 

2.555e^0.01644(t+x) = 3(2.555)e^0.01644t

e^0.01644(t+x) = 3e^0.01644t

(e^0.01644(t+x)) / (e^0.01644t) = 3

e^{0.01644(t+x) - 0.01644t} = 3

e^{0.01644t + 0.01644x - 0.01644t} = 3

e^0.01644x = 3

ln3 = 0.01644x

1.0986 = 0.01644x

66.826 = x

 

So, the tripling time is 66.826 years.