Mathematical Methods — Question 15

QCAA 2021, Paper 2 · 4 marks

Q15 · 2021 · Technology-activeComplex familiar4 marks

QUESTION 15 (4 marks)

A new internet search engine gives a ranking

R

to each website based on the function

R=\log_{10}(50h^2)

, where

h

is the number of hits (visits) the website has received. If a website currently has 100 hits, determine how many more hits they need to increase their ranking by 1.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

217 additional hits

Worked solution

**Method 1.** Current ranking

=\log_{10}(50\times100^2)=5.69897

, so the increased ranking is

6.69897

. Solving

6.69897=\log_{10}(50h^2)

gives

h=316.228

. Therefore the website requires an additional 217 hits.

**Alternative Method 2.** Write

R_{old}=\log_{10}(50)+2\log_{10}h

and let

k

be the additional hits. From

R_{new}=R_{old}+1

2\log_{10}(h+k)-2\log_{10}h=1,

\frac{h+k}{h}=\sqrt{10}

. With

h=100

k=216.228

, therefore 217 additional hits are required.

correctly determines the website’s increased ranking

[1 mark]

determines number of hits for increased ranking

[1 mark]

provides reasonable solution for number of additional hits

[1 mark]

shows logical organisation communicating key steps

[1 mark]

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Mathematical Methods — Question 15

QUESTION 15 (4 marks)

QCAA guide · typeset solution

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