Mathematical Methods — Question 17

QCAA 2024, Paper 1 · 3 marks

Q17 · 2024 · Technology-freeComplex familiar3 marks

QUESTION 17 (3 marks)

A community group that uses social media created a new post on the internet on a day when they had 1000 members. The rate of change in their number of members (members/day) is given by

f'(t)=3e^{0.5t}

, where

t

represents days after the new post. Determine the time it will take for the community group to achieve seven times the initial number of members. Express your answer in the form

a\ln(b)

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WORKED SOLUTION

QCAA guide · typeset solution

3 marks

ANSWER

2\ln(1001)

days.

Worked solution

Method 1: Seven times the initial membership is 7000, so the required increase is 6000. Let

m

be the time when 7000 members is reached. Then

6000=\int_0^m3e^{0.5t}\,dt=[6e^{0.5t}]_0^m=6e^{0.5m}-6.

Thus

6006=6e^{0.5m}

, so

e^{0.5m}=1001

and

m=2\ln(1001)\text{ days}.

Method 2: Integrating gives

f(t)=6e^{0.5t}+c.

Since

f(0)=1000

c=994

. Setting

f(m)=7000

gives

7000=6e^{0.5m}+994,

so again

m=2\ln(1001)

days.

correctly uses the initial conditions to determine the increase

[1 mark]

correctly determines the integral

[1 mark]

determines the number of days required

[1 mark]

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Mathematical Methods — Question 17

QUESTION 17 (3 marks)

QCAA guide · typeset solution

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