Question

A lake initially contains 3000 fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by 7% each month. However, factoring in all causes, 300 fish are lost each month. How many fish will be in the pond after 5 months

Answer 1

Answer:

2,482 fishes will be in the pond after 5 months

Step-by-step explanation:

First, let us model an equation that is used to determine the fish population after n months:

Let the initial population of the fishes = P₀

Let the population after n months = Pₙ

percentage increase each month = 7%

$P_n = P_{n-1} + 7 \%P_{n-1}\\P_n = P_{n-1} + 0.07P_{n-1}\\P_n = (1.07)P_{n-1}$

Next, we are told that 300 fishes are lost each month, therefore, the population after n months is given by:

$P_n = (1.07)P_{n-1} - 300$

Therefore population after 5 months:

n = 5

$P_5 = (1.07)P_{5-1} - 300\\P_5 = (1.07)P_4 - 300$

$Remember\ P_0 = 3000\\starting\ at\ P_1\\P_1 = (1.07)P_0 - 300\\P_1 = (1.07)(3000) - 300\\P_1 = 3210 - 300 = 2910\\P_2 = (1.07)P_1 - 300\\P_2 = (1.07)(2910) - 300 = 2813.7\\\\P_3 = (1.07)(2813.7) - 300\\P_3 = 3010.66 - 300 = 2710.66\\\\P_4 = (1.07)(2710.66) - 300 = 2600.40\\\\P_5 = (1.07)(2600.40) - 300 = 2482.4$

≈ 2,482 fishes