QCE Mathematical Methods: Exam practice Q12 (2024, Paper 2)

Mathematical Methods — Question 12

QCAA 2024, Paper 2 · 5 marks

Q12 · 2024 · Technology-activeSimple familiar5 marks

QUESTION 12 (5 marks)

The magnitude of an earthquake can be modelled by the logarithmic equation

M_A=\log_{10}\left(\frac{I_A}{I_0}\right)

, where

M_A

is the magnitude at a location A,

I_A

is the intensity of the earthquake at location A and

I_0

is a constant. An earthquake at location P had a magnitude of 5.2. A different earthquake at location Q had a magnitude of 3.5.

Determine an equation involving logarithms that expresses the difference in magnitudes between the earthquakes at locations P and Q.

[1 mark]

How many times more intense was the earthquake at location P than the earthquake at location Q?

[4 marks]

WORKED SOLUTION

5 marks

ANSWER

M_P-M_Q=\log_{10}(I_P/I_0)-\log_{10}(I_Q/I_0)

; b) approximately

50.1187

times.

Worked solution

a) At P,

M_P=\log_{10}\left(\frac{I_P}{I_0}\right),

and at Q,

M_Q=\log_{10}\left(\frac{I_Q}{I_0}\right).

Hence

M_P-M_Q=\log_{10}\left(\frac{I_P}{I_0}\right)-\log_{10}\left(\frac{I_Q}{I_0}\right).

b) This simplifies to

M_P-M_Q=\log_{10}\left(\frac{I_P}{I_Q}\right).

Changing to index form,

\frac{I_P}{I_Q}=10^{M_P-M_Q}=10^{1.7}=50.1187.

Therefore the earthquake at P was approximately 50.1187 times as intense as at Q.

correctly determines an equation involving the difference in magnitudes

[1 mark]

determines a simplified logarithmic equation

[1 mark]

states an equation with intensities made the subject, i.e. the log equation is converted to an index equation

[1 mark]

substitutes the given magnitude values appropriately

[1 mark]

determines how more intense the earthquake is at location P compared with location Q

[1 mark]

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

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