Question

In an experiment, a number of fruit flies are placed in a container. The population of fruit flies, P, increases and can be modelled by the function|PKA 304987/10, where t is the number of days since the fruit flies were placed in the container. a. Find the number of fruit flies which were initially placed in the container. b. Find the number of fruit flies that are in the container after 6 days. C. The maximum capacity of the container is 8,000 fruit flies. Find the number of days until the container reaches its maximum capacity.d. The number of fruit flies will not decrease below p. Write down the value of p.

Answer 1

`P(t)=12*3^(0.498t),t\ge0`

a. Find the number of fruit flies which were initially placed in the container.

Evaluate the function for t = 0

`\begin{gathered} t=0 \\ P(0)=12*3^(0.498(0)) \\ P(0)=12*3^0 \\ P(0)=12*1 \\ P(0)=12 \end{gathered}`

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b. Evaluate the function for t = 6

`\begin{gathered} t=6 \\ P(6)=12*3^(0.498(6)) \\ P(6)=12*3^(2.988) \\ P(6)=319.7566278 \\ P(6)\approx320 \end{gathered}`

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c.

Evaluate the function for P(t) = 8000

`\begin{gathered} P(t)=8000 \\ 8000=12*3^(0.498t) \\ solve_{\text{ }}for_{\text{ }}t\colon \\ (8000)/(12)=3^(0.498t) \\ \ln ((8000)/(12))=0.498t\ln (3) \\ t=(\ln ((8000)/(12)))/(0.498\ln (3)) \\ t=11.88481843 \\ t\approx12 \end{gathered}`

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d.

Let's find the limit for t->-∞ :

`\begin{gathered} \lim _(t\to\infty)P(t)=p=4\cdot3^(0.498(-\infty))=0=p \\ \end{gathered}`
`p=0`