Answer:
Step-by-step explanation:
Treat this like compound interest: Use A = P(1 + r)^t.
Here, P is the initial population and A is 3 times that, or 3P. Since P = 20 people, 3P = 60 people,
and this population is reached after 100 years.
We need to determine r, substitute its value into the formula A = P(1 + r)^t, and then determine the population of the village after 400 years.
60 = 20(1 + r)^100
Simplifying, 3 = (1 + r)^100.
Taking the natural log of both sides,
ln 3 = 100 ln (1 + r), or
ln 3
ln (1 + r) = ---------------
100
= 1.0986 / 100 = 0.01986
We must solve this for r. Raising e to the power ln (1 + r), on the left side of an equation, and raising e to the power 0. 01986 on the right side, we get:
1 + r = 3, so r must = 2.
Now find the pop of the village today. Use the same equation: A = P (1+r)^t.
A = 20(1 +2)^4 (hundreds),
or
A = 20(3)^4, or
A = 81
The population after 400 years is 81.
