Mathematical Methods — Question 14

QCAA 2023, Paper 1 · 6 marks

Q14 · 2023 · Technology-freeSimple familiar6 marks

QUESTION 14 (6 marks)

The rate of change in the number of bacteria in a science experiment is represented by

\frac{dP}{dt}=e^{2t},\qquad t\ge0,

where

t

represents the time (hours) since starting the experiment and

P

represents the number of bacteria present (thousands). Initially there are 60 000 bacteria present, i.e.

P(0)=60

Determine the equation for

P(t)

[2 marks]

Determine the change in the number of bacteria during the third hour. Express your answer in terms of

e

[2 marks]

Determine how long it will take for the number of bacteria present to double after starting the experiment.

[2 marks]

WORKED SOLUTION

6 marks

ANSWER

P(t)=\frac12e^{2t}+59.5

; b)

\frac12e^6-\frac12e^4

; c)

t=\frac12\ln121

hours.

Worked solution

P(t)=\int e^{2t}\,dt=\frac12e^{2t}+c.

Since

P(0)=60

60=\frac12e^{2(0)}+c,

c=59.5

. Therefore

P(t)=\frac12e^{2t}+59.5.

b) The total change during the third hour (

t=2

t=3

) is

P(3)-P(2)=\left(\frac12e^6+59.5\right)-\left(\frac12e^4+59.5\right)=\frac12e^6-\frac12e^4.

60\times2=120

120=\frac12e^{2t}+59.5\Rightarrow60.5=\frac12e^{2t}\Rightarrow121=e^{2t}.

Hence

t=\frac12\ln121.

correctly determines the integral from the rate

[1 mark]

determines the value of c

[1 mark]

identifies the method required for the third hour calculation

[1 mark]

determines an expression for the total change

[1 mark]

correctly determines the doubled population

[1 mark]

determines expression for the time required

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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