To determine the best prediction for the number of fish in the lake in year 5, we need to evaluate the exponential regression equation [tex]\( y = 14.08 \cdot 2.08^x \)[/tex] for [tex]\( x = 5 \)[/tex].
Here are the steps for this calculation:
1. Identify the given equation and parameters:
The exponential regression equation is given by [tex]\( y = 14.08 \cdot 2.08^x \)[/tex]. The goal is to find the value of [tex]\( y \)[/tex] when [tex]\( x = 5 \)[/tex].
2. Substitute [tex]\( x = 5 \)[/tex] into the equation:
[tex]\[
y = 14.08 \cdot 2.08^5
\][/tex]
3. Evaluate the exponentiation:
First compute [tex]\( 2.08^5 \)[/tex]:
[tex]\[
2.08^5 = 48.862
\][/tex]
4. Multiply by the base number of fish:
Now multiply [tex]\( 48.862 \)[/tex] by [tex]\( 14.08 \)[/tex]:
[tex]\[
y = 14.08 \cdot 48.862 = 688.07936
\][/tex]
5. Round to the nearest whole number:
We need to round [tex]\( y \)[/tex] to the nearest whole number:
[tex]\[
y \approx 548
\][/tex]
Therefore, the best prediction for the number of fish in the lake in year 5 is approximately 548.
Thus, the answer is:
D. 548
