A biologist models the populations of two fish species, Species A and Species B, in a lake over time. At time t = 0 mon…

To find an expression for k, set up the linear equations for the two population models. At time t, the population of Species A is given by P_A - r_A · t, and the population of Species B is given by P_B + r_B · t. The populations are equal at t = t_1, so P_A - r_A · t_1 = P_B + r_B · t_1, which rearranges to P_A - P_B = (r_A + r_B) · t_1. In the modified scenario, the rate of increase for Species B is r_B · (1 + k / 100). The populations are equal at t = t_2, so P_A - r_A · t_2 = P_B + r_B · (1 + k / 100) · t_2, which rearranges to P_A - P_B = [r_A + r_B + r_B · (k / 100)] · t_2. Since the initial population difference P_A - P_B is the same in both scenarios, equate the two expressions: (r_A + r_B) · t_1 = (r_A + r_B) · t_2 + r_B · (k / 100) · t_2. Subtracting (r_A + r_B) · t_2 from both sides yields (r_A + r_B) · (t_1 - t_2) = (k / 100) · r_B · t_2. Multiplying both sides by 100 and dividing by r_B · t_2 gives k = (100(r_A + r_B)(t_1 - t_2)) / (r_B · t_2).

Choice B is incorrect because it uses t_1 in the denominator instead of t_2, incorrectly using the initial time to isolate the rate change.

Choice C is incorrect because it inverts the ratio between r_B and the combined rate (r_A + r_B).

Choice D is incorrect because it treats the percent change in the rate as identical to the relative change in time, ignoring the contributions of r_A and r_B.