The population of California was 29.76 million in 1990 and 33.87 million in 2000. Assume that the population grows exponentially.

Let $A}_{0$ and r be the initial population & growth rate of population respectively. Then population in California t years after 1990 is,

$P\left(t\right)={A}_{0}{e}^{rt}$ (1)

a. Since the population in 1990 is 29.76 million, therefore ${A}_{0}=29.76$

In 2000, the population is 33.87 million. That is,

$P\left(t\right)=33.87$ when $t=10$

Therefore from equation (1), we get

$33.87=29.76{e}^{10r}$

$\Rightarrow {e}^{10r}=1.13810483$

$\Rightarrow 10r=\mathrm{ln}\left(1.13810483\right)$

$\Rightarrow r=\frac{\mathrm{ln}\left(1.13810483\right)}{10}$

$\Rightarrow r=0.012936$

Therefore the population t years after 1990 is,

$P\left(t\right)=29.76{e}^{0.012936t}$

b) The double of 29.76 million is 59.52 million.

So put $P(t)=59.52$ in $P\left(t\right)=29.76{e}^{0.012936t}$, we get

$59.52=29.76{e}^{0.012936t}$

$\Rightarrow {e}^{0.012936t}=2$

$\Rightarrow 0.012936t=\mathrm{ln}\left(2\right)$

$\Rightarrow t=\frac{\mathrm{ln}\left(2\right)}{0.012936}$

$\Rightarrow t=53.5828$

Hence the population will double in 53.5828 years.

c)Since 2010−1990=20, therefore to predict the population in 2010, put $t=20$ in $P\left(t\right)=29.76{e}^{0.012936t}$ , we get

$P\left(20\right)=29.76{e}^{0.012936\left(20\right)}$

$=38.5472$ million

The population in California in 2010 is 37.3 million, therefore the actual population is approximately 1 million less than the obtained population.