Question

A population \[P(t)\]of 1000 bacteria introduced into a nutrient medium grows according to \[P(t) = 1000 + \frac{{1000t}}{{100 + {t^2}}}\]. Then the maximum size of the bacterial population is
A. 1100
B. 1250
C. 1050
D. 5250

Answer 1

Hint: Bacteria are tiny single-celled organisms. Bacteria have been present almost anywhere on earth and are essential to the planet's environments. Some species will be able to live in the most extreme conditions of temperature and pressure. The human body is full of bacteria, and in fact, must contain more bacterial cells than human cells.

Complete step-by-step answer:
Bacteria are unicellular organisms that tend to replicate asexually by the method of binary fission. The growth of bacterial cultures is described as a rise in the number of bacteria in a population instead of in the size of individual cells. The growth of these bacterial cells occurs in an exponential manner, i.e., one cell is split into 2, then 4, then 8, 16, 32, and so on. At present, there are 1000 bacteria, after 1 hour there would be 500 because the number of bacteria doubles each and every hour. After 2 hours there would be half of 500 bacteria, that is 250 bacteria. After 3 hours there would be 125 bacteria. The time required for a bacterial cell to double is known as generation time. The generation time differs among varied species of bacteria based upon the environmental conditions they are growing.

Given,
\[P(t) = 1000 + \frac{{1000t}}{{100 + {t^2}}}\]
\[\frac{{dp}}{{dt}}\]\[\frac{{dp}}{{dt}}\]for maximisation
=>\[\frac{{[0 + (100 + {t^2})*1000 - 1000t(2t)]}}{{{{[100 + {t^2}]}^2}}} = 0\]
=>\[1000[100 + {t^2} - 2{t^2}] = 0\]
=>\[100 - {t^2} = 0\]
=>\[t = \pm 10\]
Maximal will be at \[t = \pm 10\]
\[p(t) = \frac{{1000 + [1000*10]}}{{[100 + 100]}}\]
\[p(t) = 1000 + [\frac{{10000}}{{200}}]\]
\[\begin{array}{l}p(t) = 1000 + 50\\p(t) = 1050\end{array}\]

Therefore the correct answer is option C.

Note: One of the first-ever organisms evolving on earth was thought to be a unicellular organism, similar to contemporary bacteria. Currently, bacteria are treated as one of the most ancient forms of life on earth. Although most bacteria cause us to have a disease, they have long-term, mutual relations with humans and are very much significant to our survival.