#### Answer

a) The exponential growth function is $ A=679{{e}^{0.00235t}}$
b) The population of Europe will be 800 million in the year $2045$

#### Work Step by Step

(a)
Let us consider the exponential growth function as shown below:
$ A={{A}_{o}}{{e}^{kt}}$ (I)
Here, $ t $ is the number of years after 1975.
In 1975, the population of Europe was 679 million.
So, for $ t=0,\ A=679.$
Put the above values in equation (1):
$\begin{align}
& A={{A}_{o}}{{e}^{kt}} \\
& 679={{A}_{o}}{{e}^{k\times 0}} \\
& {{A}_{o}}=679
\end{align}$
Substitute the value of ${{A}_{o}}$ in equation (I).
So, the exponential growth function after putting the value of ${{A}_{0}}$ is as shown below:
$ A=679{{e}^{kt}}$ (II)
In the year 2015, the population of Europe was 746 million.
So
$\begin{align}
& t=2015-1975 \\
& =40
\end{align}$
And
$ A=746$
Put the values of $ A $ and $ t $ in equation (II), and simplify as shown below:
$\begin{align}
& 746=679{{e}^{k\times 40}} \\
& {{e}^{k\times 40}}=\frac{746}{679} \\
& \ln \left( {{e}^{k\times 40}} \right)=\ln \left( \frac{746}{679} \right) \\
& 40k\times \ln \ e=\ln \left( \frac{746}{679} \right) \\
\end{align}$
Simplify it further:
$\begin{align}
& 40k\times 1=0.094104 \\
& k=\frac{0.094104}{40} \\
& =0.002352 \\
& \approx 0.00235
\end{align}$
Put the value of $ k $ in equation (II), so the exponential growth function is
$ A=679{{e}^{0.00235t}}$
(b)
In the above part (a), the exponential growth function is as shown below:
$ A=679{{e}^{0.00235t}}$
It is stated in the problem that $ A=800.$
Put the value of $ A $ in the above equation, and simplify as shown below:
$\begin{align}
& 800=679{{e}^{0.00235t}} \\
& {{e}^{0.00235t}}=\frac{800}{679} \\
& \ln \left( {{e}^{0.00235t}} \right)=\ln \left( \frac{800}{679} \right) \\
& 0.00235t=0.16399
\end{align}$
$\begin{align}
& t=\frac{0.16399}{0.00235} \\
& =69.7832 \\
& \approx 70
\end{align}$
Hence, the required year will be $1975+70=2045$ in which the population of Europe will be 800 million.