a) An exponential model will be of the form #p = a xx r^n#, where the population after #n# years is #p# and the original population is #a#. The rate at which it increases (in decimals) is #r#.

Our equation, therefore is #p = 20,000,000(1.02)^n#

b) To find the population after #2016#, we need to insert #n = 4#, because the time between the end of #2012# and the end of #2016# is #4# years.

#p = 20,000,000(1.02)^4#

#p = 21,648,643#

Hence, the population at the end of #2016# will be #21,648,643#.

c) Set #p# to #25,000,000# and solve.

#p = 20,000,000(1.02)^n#

#25,000,000= 20,000,000(1.02)^n#

#1.25 = (1.02)^n#

#ln(1.25) = n(ln1.02)#

#n = ln(1.25)/ln(1.02)#

#n ~=11.268#

However, we can't round to #11# since after #11# years, we won't have attained #25,000,000# yet. After 12 years we will have, though. So, in the year #2012 + 12#, or #2024#, Nigeria will obtain a population of #25,000,000#

d) The doubling population will be #40,000,000# (since #20,000,000 xx 2 = 40,000,000)#

#40,000,000 = 20,000,000(1.02)^n#

#2 = (1.02)^n#

#ln2 = nln(1.02)#

#n = ln2/ln(1.02)#

#n = 35.003#

Again, as with part c), we're going to have to use #n = 36# since when #n = 35#, the population will not have yet doubled.

After #36# years, we will be in #2012 + 36 = 2048#.

Hopefully this helps!