1. ## Optimization

I really need help understanding optimization, so if someone could explain it to me, i'd be grateful.

The number of bacteria in a culture at time t is given by
N= 1000(25+te to the -t/20)
for 0 is less than or equal to t, which is less than or equal to 100.

a) Find the largest and the smallest number of bacteria in the culture during the interval.
b) At what time during the interval is the rate of change in the number of bacteria a minimum?

Thanks.

2. Originally Posted by turtle
I really need help understanding optimization, so if someone could explain it to me, i'd be grateful.

The number of bacteria in a culture at time t is given by
N= 1000(25+te to the -t/20)
for 0 is less than or equal to t, which is less than or equal to 100.

a) Find the largest and the smallest number of bacteria in the culture during the interval.
b) At what time during the interval is the rate of change in the number of bacteria a minimum?

Thanks.
First sketch the graph - see attachment.

a) From the graph we see that the smallest population occurs when $t=0$,
which is not a surprise as at all other times $N$ is larger by $1000 t e^{-t/20}>0$.

the maximum is of smooth type and so we finc it by differentialtion $N$ with respect
to t, and setting the derivative to zero and solving the resulting equation:

$
\frac{d}{dt} 1000(25+ te^{-t/20} ) = 1000[e^{-t/20}-\frac{t}{20} e^{-t/20}]
$

which is zero when:

$
1-\frac{t}{20}=0
$

or $t=20$, when the population is: $N \approx 32357.6$

RonL