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A population p(t) of 1000 bacteria inrod...

A population `p(t)` of 1000 bacteria inroduced into nutrient medium grows according to the relation `p(t)=1000+(1000t)/(100+t^2)`. The maximum size of this bacterial population is

A

1100

B

1250

C

1050

D

5250

Text Solution

Verified by Experts

The correct Answer is:
C
`p(t)=1000+(1000t)/(100+t^(2))`
`(dp)/(dt)=((100+t^(2))1000-1000t.2t)/((100+t^(2))=(100(100-t^(2)))/((100+t^(2)))`
For extremum,`(dp)/(dt)=0impliest=10`
Now `(dp)/(dt)|_(tlt10)gt0` and `(dp)/(dt)|_(tgt10)lt0`
At `t=10,(dp)/(dt)` change from positive to negative.
p is maximum at `t=10`.
`p_(max)=p(10)=1000+(1000.10)/(100+10^(2))=1050`
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