A microbiology experiment monitors the number of bacteria,

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

B(t)

, in a culture for

0\le t \le 8

hours. A graph of

\ln B

versus

t

(with

\ln B

on the vertical axis and

t

in hours on the horizontal axis) is described as follows: • The graph is a straight line with slope

0.28

for

0\le t<4

. • At

t=4

the line changes suddenly; for

t>4

the graph is a straight line with slope

0.12

. • The point

(4,\ln 6000)

lies on the graph. Assume

B(t)

is differentiable for all

t\ne4

and that

\dfrac{dB}{dt}

is proportional to

B

on each open interval

(0,4)

and

(4,8)

. (a) Describe what the two distinct straight‐line segments of the

\ln B

graph reveal about the behavior of

\dfrac{1}{B}\dfrac{dB}{dt}

on the intervals

0<t<4

and

4<t<8

. Include an explanation of why this behavior implies different constant values of proportionality. (b) Explain, without performing explicit calculations, whether the bacterial population is growing more rapidly (in terms of percentage increase per hour) before or after

t=4

. Justify your conclusion using features of the graph. (c) Let

k_1

and

k_2

denote the constant values of

\dfrac{1}{B}\dfrac{dB}{dt}

on the intervals

0<t<4

and

4<t<8

, respectively. Use the given point on the

\ln B

graph to relate

k_1

and

k_2

to the numerical value

6000

and interpret the units of

k_1

and

k_2

in this context.

Related theory

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