Mathematical Methods — Question 17

QCAA 2020, Paper 1 · 6 marks

Q17 · 2020 · Technology-freeComplex familiar6 marks

QUESTION 17 (6 marks)

The volume of water in a tank is represented by a function of the form

V(t)=Ae^{kt},

where

V

is in litres and

t

is in minutes. Initially, the volume is 100 litres and it is decreasing by 50 litres per minute. Determine the time at which the volume is decreasing at the rate of

\frac{50}{7}

litres per minute. Express your answer in the form

\ln(a)

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

t=\ln(49)

minutes

Worked solution

t=0

V=100

, so

A=100

and

V(t)=100e^{kt}

. Then

V'(t)=100ke^{kt}

. Since

V'(0)=-50

k=-1/2

. Thus

V'(t)=-50e^{-0.5t}

. Set

V'(t)=-50/7

\frac17=e^{-0.5t}.

Hence

t=-2\ln(1/7)=\ln(49)

correctly determines

A

[1 mark]

determines

k

[1 mark]

establishes equation in

t

[1 mark]

determines

t

[1 mark]

determines

t

in required form

[1 mark]

shows logical organisation communicating key steps

[1 mark]

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Mathematical Methods — Question 17

QUESTION 17 (6 marks)

QCAA guide · typeset solution

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