QCE Mathematical Methods: Exam practice Q14 (2022, Paper 1)

Mathematical Methods — Question 14

QCAA 2022, Paper 1 · 6 marks

Q14 · 2022 · Technology-freeSimple familiar6 marks

QUESTION 14 (6 marks)

The rate that water fills an empty vessel is given by

\dfrac{dV}{dt}=0.25e^{0.25t}

(in litres per hour),

0\le t\le8\ln(6)

, where

t

is time (in hours).

Determine the function that represents the volume of water in the vessel (in litres).

[2 marks]

The vessel is full when

t=8\ln(6)

. Determine the volume of water, to the nearest litre, the vessel can hold when full.

[2 marks]

The table shows the approximate rate the water flows into the vessel at certain times.

Use information from the table and the trapezoidal rule to determine the approximate volume of water in the vessel after three hours.

[2 marks]

WORKED SOLUTION

6 marks

ANSWER

V(t)=e^{0.25t}-1

; b) 35 litres; c) 1.12 litres.

Worked solution

V=\int0.25e^{0.25t}\,dt=e^{0.25t}+c.

Since the vessel is empty at

t=0

0=1+c

, so

c=-1

and

V(t)=e^{0.25t}-1.

V(8\ln6)=e^{2\ln6}-1=36-1=35\text{ litres}.

c) Using the trapezoidal rule,

V\approx\frac12[0.25+0.53+2(0.32+0.41)]=1.12\text{ litres}.

correctly determines the integral of the function

V(t)

[1 mark]

determines the value of

c

[1 mark]

determines the simplified exponential term

[1 mark]

determines number of litres

[1 mark]

establishes expression for approximate number of litres of water in vessel after 3 hours

[1 mark]

determines approximate number of litres

[1 mark]

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

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