To project the gray wolf population of the Western Great Lakes in 2025 using the exponential function [tex]\( f(x) = 1145 e^{0.0325x} \)[/tex], let's go through the steps carefully.
### Step-by-Step Calculation:
1. Determine the number of years after 1978 that corresponds to 2025:
[tex]\[
\text{Years after 1978} = 2025 - 1978 = 47
\][/tex]
2. Evaluate the population function [tex]\( f(x) \)[/tex] at [tex]\( x = 47 \)[/tex]:
[tex]\[
f(47) = 1145 \cdot e^{0.0325 \cdot 47}
\][/tex]
3. Calculate [tex]\( e^{0.0325 \cdot 47} \)[/tex] (approximated here for clarity):
Let's denote [tex]\( 0.0325 \cdot 47 = 1.5275 \)[/tex], then:
[tex]\[
e^{1.5275} \approx 4.606064
\][/tex]
4. Multiply by the initial population coefficient:
[tex]\[
f(47) = 1145 \cdot 4.606064 \approx 5275
\][/tex]
5. Round to the nearest whole number:
Since the function's output [tex]\( 5275 \)[/tex] is already a whole number, we don't need further rounding.
### Conclusion:
The projected gray wolf population of the Western Great Lakes in 2025, rounded to the nearest whole number, is 5275.
Thus, the correct option is:
b. 5275
