The formula P = 0.64x^2 - 0.044x + 3 models the approximate population P, in thousands, for a species of fish in a local pond, x years after 1997. During what year will the population reach 34,052 fish?

To find the year when the population reaches 34,052 fish, we need to solve the equation 0.64x^2 - 0.044x + 3 = 34.052.

Step 1: Rewrite the equation
0.64x^2 - 0.044x + 3 = 34.052

Step 2: Subtract 34.052 from both sides
0.64x^2 - 0.044x - 31.052 = 0

Step 3: Use the quadratic formula
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).

In this case, a = 0.64, b = -0.044, and c = -31.052.

Using these values in the quadratic formula, we can find the values of x.

x = (-(-0.044) ± √((-0.044)^2 - 4*(0.64)*(-31.052))) / (2*(0.64))

Simplifying further,

x = (0.044 ± √(0.001936 + 79.2896)) / (1.28)
x = (0.044 ± √(79.291536)) / (1.28)
x = (0.044 ± 8.907) / (1.28)

Now, we have two possible values for x.

x1 = (0.044 + 8.907) / 1.28
x2 = (0.044 - 8.907) / 1.28

Calculating these values,

x1 = 8.951 / 1.28 = 6.9984375
x2 = -8.863 / 1.28 = -6.91015625

Step 4: Convert x to the corresponding year
We know that x represents the number of years after 1997. Since x cannot be negative in this case, we discard the negative value (-6.91015625).

To find the corresponding year, we add x to 1997:

Year = 1997 + x1
Year = 1997 + 6.9984375
Year ≈ 2004.998

Therefore, the population will reach 34,052 fish in approximately the year 2005 (option d).