Question

A species of fish was added to a lake. The population size P (t) of this species can be modeled by the following function, where t is the number of years from the time the species were added to the lake.Find the population size of the species after 4 years and the population size after 8 yearsRound your answers to the nearest whole number as necessary.

Answer 1

a) 223 fishes

b) 352 fishes

Step-by-step explanation:
`\begin{gathered} \text{The given function:} \\ P(t)\text{ = }(1200)/(1+8e^(-0.15t)) \\ \\ We\text{ n}eed\text{ to find the }P(t)\text{ when t = 4 years and t = 8 years} \end{gathered}`
`\begin{gathered} \text{when t = 4 years} \\ \text{substitute 4 for t in the given function} \\ P(t)\text{ = }(1200)/(1+8e^(-0.15(4))) \\ P(t)\text{ = }(1200)/(1+4.3905)\text{= }(1200)/(5.3905) \\ P(t)\text{ = }222.61 \\ \\ To\text{ the nearest whole number, population size is 223 fishes} \end{gathered}`
`\begin{gathered} \text{when t = 8 years} \\ P(t)\text{ = }(1200)/(1+8e^(-0.15(8))) \\ P(t)\text{ =}(1200)/(1+2.4096)\text{ = }(1200)/(3.4096) \\ P(t)\text{ = }351.95 \\ \\ To\text{ the nearest whole number, population size is 352 fishes} \end{gathered}`