Question

The population N(t) (in millions) of a country t years after 1980 may be approximated by the formula N(t) = 213e^0.0101t. When will the population be twice what it was in 1980?

Answer 1

Population t years after 1980 is given by:

𝒹ₚ/𝒹ₜ = N(t) = 213 e⁰·⁰¹⁰¹ ᵗ

integrating it we get,
P= 213·C·e⁰·⁰¹⁰¹ ᵗ ...(i)

[Where C is a constant of integration]

when t=0 i.e. in 1980,
P= 213C
when population of 1980 is doubled,
i.e P= 2P= 2×213·C

Now from equation (i)
2×213·C = 213·C·e⁰·⁰¹⁰¹ ᵗ

2=e⁰·⁰¹⁰¹ ᵗ
taking log on both sides,
log 2 = 0.0101 t
t = (log 2)/0.0101= 29.54 years

Answer 2

what a lot of work, when you just want to find when
e^0.0101t = 2

Answer 3

Actually this was all to explain the person who posted the problem.
I could have skipped some lines.
I don't know what his/her level of education is, so I just used my own method to solve it and also explained it well enough to be understood.
Thank you for your suggestion.