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

Respuesta :

Answer:

2002.7 (1 d.p.).

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

Step-by-step explanation:

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

[tex]f(x)=1.8208e^{0.3387x}[/tex]

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.

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

Add the found value of x to the year 1998:

[tex]\implies 1998+4.65153751...=2002.65153...[/tex]

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.).

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