Mathematical Methods — Question 19

QCAA 2023, Paper 1 · 7 marks

Q19 · 2023 · Technology-freeComplex unfamiliar7 marks

QUESTION 19 (7 marks)

Jaxon and Shari each own a shop and have recorded the number of customers entering their shop on two consecutive days.

	Day 1	Day 2
Jaxon’s customers	40	30
Shari’s customers	10	20
Total	50	50

The number of daily customers for each shop can be modelled by the equation

y=A\ln(Bx)

, where

x

is the day and

y

is the number of customers. The constants

A

and

B

are different for each shop. Determine algebraically whether the total number of customers for Jaxon and Shari’s shops will be the same every day in the future.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

Yes. The sum of the two models is

50

for every day

x

Worked solution

For each shop use

y=A\ln(Bx)

as the model.

**Shari’s shop**

10=A\ln B\qquad(1)

20=A\ln(2B)\qquad(2)

20=A\ln2+A\ln B=A\ln2+10

, hence

A=\frac{10}{\ln2}.

Substituting into (1) gives

\ln B=\ln2

, so

B=2

. Therefore

y=\frac{10}{\ln2}\ln(2x).

**Jaxon’s shop**

40=A\ln B\qquad(1)

30=A\ln(2B)\qquad(2)

30=A\ln2+40

, hence

A=-\frac{10}{\ln2}.

Substituting into (1) gives

\ln B=-4\ln2=\ln(2^{-4})

, so

B=2^{-4}=\frac1{16}.

Therefore

y=-\frac{10}{\ln2}\ln\left(\frac{x}{16}\right).

On any day

x

, the sum is

\frac{10}{\ln2}\ln(2x)-\frac{10}{\ln2}\ln\left(\frac{x}{16}\right)=\frac{10}{\ln2}\ln(32)=50.

The sum of the models therefore does result in the same number of customers, i.e. 50 on any day.

correctly determines A (or B) for Shari’s shop

[1 mark]

determines the model for Shari’s shop

[1 mark]

correctly determines A (or B) for Jaxon’s shop

[1 mark]

determines the model for Jaxon’s shop

[1 mark]

determines an expression for the total number of daily customers in both shops

[1 mark]

justifies the sum obtained by explaining mathematical reasoning

[1 mark]

provides an appraisal by interpreting the result of the analysis of the sum of the two models performed

[1 mark]

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Mathematical Methods — Question 19

QUESTION 19 (7 marks)

QCAA guide · typeset solution

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