Question

The value of China's exports of automobiles and parts (in billions of dollars) is approximately f(x)=1.8208e^.3387x, where x = 0 corresponds to 1998.

In what year did/will the exports reach $8.8 billion?



Give your answer as the year, with at least one decimal place

Answer 1

2002.7 (1 d.p.).

The value of the exports reached $8.8 billion during the year 2002.

Explanation:

The function modelling the approximate value of China's exports of automobiles and parts (in billions of dollars) is:

`f(x)=1.8208e^(0.3387x)`

where x = 0 corresponds to 1998.

To find the year when the exports reach $8.8 billion, substitute f(x) = 8.8 into the given equation, solve for x, then add this value to 1998.

`\begin{aligned}1.8208e^(0.3387x)&=8.8\\\\e^(0.3387x)&=(8.8)/(1.8208)\\\\\ln e^(0.3387x)&=\ln \left((8.8)/(1.8208)\right)\\\\0.3387x&=\ln \left((8.8)/(1.8208)\right)\\\\x&=(\ln \left((8.8)/(1.8208)\right))/(0.3387)\\\\x&=4.65153751...\end{aligned}`

Add the found value of x to the year 1998:

`\implies 1998+4.65153751...=2002.65153...`

Therefore, the the value of the exports reached $8.8 billion during the year 2002.

As the answer requires the year with at least one decimal place, the final answer is 2002.7 (1 d.p.).