a) n(t)=n02(t/a) now at time t=0, n=n0 but at t=19 years, n=2n0
n(t=19 years) = n0(2)(19/a) which must be equal at that time to 2n0
so n0(2)(19/a) = 2n0 so 219/a = 2 so 19/a = 1 so a = 19 years
and n(t) = n0 2t/19
b) n(t) = n0e(rt) to find what r, the coefficient of t in the exponent is,
n(t=19 years)= n0er(19 years) =2n0
so er(19 years) = 2
we take the NATURAL logarithm of both sides (that the logarithm for base e)
r(19) = ln(2)
r = ln(2)/19 = 0.0365 year-1
so n(t) = n0e(rt) = n0e,0365t
c) Sorry I don't have the graph capability.
d) what time does it take to get for the population to reach 500,000?
You can use either of the two expressions; I'll use the one from a)
n(t) = n0 2t/19= 500000
so (120000) 2t/19= 500000
2t/19 = (500000/120000)=4.17
take the natural log of both sides
ln(2t/19) = ln(4.17)
Note that ln(ab) = b ln(a) so
(t/19)ln(2) = ln(4.17)
t= 19 ln(4.17)/ln(2) ≅39.1 years (as a check notice that since the population doubles every 19 years, it will quadruple in twice that time 38 years, at which time the population with be 480000, so it makes sense it would reach 500000 a relatively short time after that.