Question

In
\[1950\], the per capita gross domestic product (GDP) of Australia was approximately
\[\$1800\]. Each year afterwards, the per capita GDP increased by approximately
\[6.7\%\].
Write a function that gives the approximate per capita GDP
\[G(t)\] of Australia
\[t\] years after
\[1950\].
Do not enter commas in your answer.
\[G(t)=\]

Answer 1

To model the per capita GDP \( G(t) \) of Australia \( t \) years after 1950 with an annual increase of approximately 6.7%, we can use the formula for exponential growth:

\[
G(t) = G_0 \cdot (1 + r)^t
\]

where:
- \( G_0 \) is the initial value (in this case, the GDP in 1950),
- \( r \) is the growth rate (expressed as a decimal), and
- \( t \) is the number of years after 1950.

Given:
- \( G_0 = 1800 \)
- \( r = 6.7\% = 0.067 \)

So, the function becomes:

\[
G(t) = 1800 \cdot (1 + 0.067)^t
\]

Now, we can simplify it:

\[
G(t) = 1800 \cdot (1.067)^t
\]

Thus, the final function for the approximate per capita GDP \( G(t) \) of Australia \( t \) years after 1950 is:

\[
\boxed{1800(1.067)^t}
\]