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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.

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Write exponential functions: word problems

Full solution

Q. 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.

Identify Values: Identify the initial value $a$ and the growth rate $r$ . The initial value $a$ is the per capita GDP in $1950$ , which is $\$1800$ . The growth rate $r$ is the annual increase in GDP, which is $6.7\%$ or $0.067$ in decimal form.
Convert Growth Rate: Convert the growth rate to a growth factor $(b)$ . The growth factor $(b)$ is calculated by adding $1$ to the growth rate. $b = 1 + r$ $b = 1 + 0.067$ $b = 1.067$
Write Exponential Function: Write the exponential function using the initial value and the growth factor. $\newline$ The function $G(t)$ that gives the approximate per capita GDP of Australia $t$ years after $1950$ is in the form $G(t) = a(b)^t$ . $\newline$ Substitute $\$1800$ for 'a' and $1.067$ for 'b' into the equation. $\newline$ $G(t) = 1800(1.067)^t$

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