babeeb0oL

2021-09-11

A fruit fly population of 24 flies is in a closed container. The number of flies grows exponentially, reaching 384 in 18 days. Find the doubling time (time for the population to double) and write an equation that models this scenario.

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Expert

As we know,
Exponential growth of population is given by
$A=P{e}^{rt}$ (1)
Here A is the population after t days, P is the initial population and r is the growth rate.
As given,
$p=24$
According to question,
Population becomes 384 in 18 days
Implies,
A=384
t=18 days
Equation becomes
$384=24{e}^{18r}$
$⇒{e}^{18r}=\frac{384}{24}$
Applying ln both sides
$18r=\mathrm{ln}\left(384\right)-\mathrm{ln}\left(24\right)$
$18r-5.951-3.178$
$⇒=0.1541$
Put the value into equation (1)
$A=P{e}^{0.1541t}$
We have to find the doubling time
That is, put A=2P
$2P=P{e}^{0.1541t}$
$⇒2={e}^{0.1541t}$
Applying ln voth sides
$\mathrm{ln}2=0.1541t$
$⇒t=\frac{\mathrm{ln}2}{0.1541}$
$=\frac{0.6931}{0.1541}$
$=4.4977$
$\approx 4$
Hence the required doubling time is 4 days.

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