A bacterial culture grows exponentially while incubated at

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

37\,^\circ\text{C}

, doubling every

3\,\text{hours}

. After

6\,\text{hours}

, the culture is transferred to a cooler environment at

25\,^\circ\text{C}

, where it continues to grow exponentially but with a new, lower growth constant. In the cooler environment, the culture doubles every

5\,\text{hours}

. Let

P(t)

denote the number of bacteria

t

hours after the culture is first placed in the

37\,^\circ\text{C}

incubator. At time

t=0

there are exactly 1,000 bacteria. (a) Determine the exact growth constants

k_1

(for

0\,\le t\le 6

) and

k_2

(for

t\ge 6

) corresponding to the two environments. (b) Write a piecewise–defined expression for

P(t)

valid for all

t\ge 0

. (c) Calculate the number of bacteria present

12\,\text{hours}

after the culture was first placed in the

37\,^\circ\text{C}

incubator. Provide an exact expression and a decimal approximation to the nearest whole bacterium. (d) An antibiotic introduced at

t=6\,\text{hours}

is observed to reduce the growth constant in the cooler environment by 30\%. Compute the new doubling time under the antibiotic treatment, expressing the answer exactly in terms of natural logarithms.

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