The value of China's exports of automobiles and parts (in billions of dollars) is approximately \( f(x)=1.8208 e^{0.3387 x} \), where \( x=0 \) corresponds to 1998. In what year did/will the exports reach \( \$ 10.6 \) billion?

To find the year when China's exports of automobiles and parts reach \( \$10.6 \) billion, we need to solve the equation: \[ f(x) = 10.6 \] Given the function: \[ f(x) = 1.8208 e^{0.3387 x} \] we set up the equation: \[ 1.8208 e^{0.3387 x} = 10.6 \] First, we divide both sides by \( 1.8208 \): \[ e^{0.3387 x} = \frac{10.6}{1.8208} \] Calculating the right side: \[ \frac{10.6}{1.8208} \approx 5.83 \] Now we have: \[ e^{0.3387 x} \approx 5.83 \] Next, we take the natural logarithm of both sides: \[ 0.3387 x = \ln(5.83) \] Calculating \( \ln(5.83) \): \[ \ln(5.83) \approx 1.653 \] Now we can solve for \( x \): \[ x = \frac{1.653}{0.3387} \approx 4.88 \] Since \( x = 0 \) corresponds to the year 1998, we find the year by adding \( x \) to 1998: \[ 1998 + 4.88 \approx 2003.88 \] This means the exports reached \( \$10.6 \) billion sometime in the year 2004. Thus, the exports reached \( \$10.6 \) billion in the year **2004**.