Question

In​ 2000, the population of a country was approximately 5.77 million and by 2077 it is projected to grow to 13 million. Use the exponential growth model $l_A = A_oe^{kt}$​, in which t is the number of years after 2000 and $A_0$ is in​ millions, to find an exponential growth function that models the data.
In which year will Israel's population be 12 million?

Answer 1

Answer:

Part 1 : $I_A = 5.77 e^{0.01055t}$

Part 2 : In 2069 the population would be 12 millions.

Step-by-step explanation:

Part 1 : Given function that shows the population( in millions ) of Israel after t years since 2000,

$I_A = A_0 e^{kt}$

If t = 0,

$I_A = 5.77$

$\implies 5.77 = A_0 e^{0}\implies A_0 = 5.77$

If t = 77 years,

The population in 2077,

$I_A = A_0 e^{77k}=5.77 e^{77k}$

According to the question,

Population in 2077 = 13 millions

$13 = 5.77 e^{77k}$

$\frac{13}{5.77} = e^{77k}$

Taking ln both sides,

$\ln(\frac{13}{5.77}) = \ln(e^{77k})$

$\ln(\frac{13}{5.77}) = 77k$

$\implies k = 0.010549\approx 0.01055$

Hence, the required function would be,

$I_A = 5.77 e^{0.01055t}$

Part 2 : If $I_A = 12$

$12 = 5.77 e^{0.01055t}$

$\frac{12}{5.77} = e^{0.01055t}$

Taking ln both sides,

$\ln(\frac{12}{5.77}) = \ln(e^{0.01055t})$

$\ln(\frac{12}{5.77}) =0.01055t$

$\implies t\approx 69$

∵ 2000 + 69 = 2069

Hence, in 2069 the population would be 12 millions.