The equation for determining exponential population growth is expressed as
A = Aox^(t/T)
where
A represents the population after time t
Ao represents the initial population
x represents the growth rate
t represents the time
T represents the time taken to achieve the change or doubling time
From the information given,
Ao = 3200
T = 16
x = 2 because it is doubling
A = 3200 * 4 = 12800 because quadruple means 4 times
By substituting these values into the formula, we have
12800 = 3200 * 2^(t/16)
Dividing both sides by 3200, we have
12800/3200 = 3200/3200 * 2^(t/16)
4 = 2^(t/16)
Taking natural log of both sides of the equation, we have
ln 4 = ln2^(t/16)
By applying one of the rules of logarithm, lna^b = blna, we have
ln 4 = (t/16)ln2
Dividing both sides of the equation by ln2, we have
ln4/ln2 = (t/16)ln2/ln2
2 = t/16
By crossmultiplying,
t = 16 * 2
t = 32
It will take 32 years