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Date May 2017 Marks available 2 Reference code 17M.2.sl.TZ1.8
Level SL only Paper 2 Time zone TZ1
Command term Find Question number 8 Adapted from N/A

Question

At Grande Anse Beach the height of the water in metres is modelled by the function \(h(t) = p\cos (q \times t) + r\), where \(t\) is the number of hours after 21:00 hours on 10 December 2017. The following diagram shows the graph of \(h\) , for \(0 \leqslant t \leqslant 72\).

M17/5/MATME/SP2/ENG/TZ1/08

The point \({\text{A}}(6.25,{\text{ }}0.6)\) represents the first low tide and \({\text{B}}(12.5,{\text{ }}1.5)\) represents the next high tide.

How much time is there between the first low tide and the next high tide?

[2]
a.i.

Find the difference in height between low tide and high tide.

[2]
a.ii.

Find the value of \(p\);

[2]
b.i.

Find the value of \(q\);

[3]
b.ii.

Find the value of \(r\).

[2]
b.iii.

There are two high tides on 12 December 2017. At what time does the second high tide occur?

[3]
c.

Markscheme

attempt to find the difference of \(x\)-values of A and B     (M1)

eg\(\,\,\,\,\,\)\(6.25 - 12.5{\text{ }}\)

6.25 (hours), (6 hours 15 minutes)     A1     N2

[2 marks]

a.i.

attempt to find the difference of \(y\)-values of A and B     (M1)

eg\(\,\,\,\,\,\)\(1.5 - 0.6\)

\(0.9{\text{ (m)}}\)     A1     N2

[2 marks]

a.ii.

valid approach     (M1)

eg\(\,\,\,\,\,\)\(\frac{{{\text{max}} - {\text{min}}}}{2},{\text{ }}0.9 \div 2\)

\(p = 0.45\)     A1     N2

[2 marks]

b.i.

METHOD 1

period \( = 12.5\) (seen anywhere)     (A1)

valid approach (seen anywhere)     (M1)

eg\(\,\,\,\,\,\)\({\text{period}} = \frac{{2\pi }}{b},{\text{ }}q = \frac{{2\pi }}{{{\text{period}}}},{\text{ }}\frac{{2\pi }}{{12.5}}\)

0.502654

\(q = \frac{{4\pi }}{{25}},{\text{ 0.503 }}\left( {{\text{or }} - \frac{{4\pi }}{{25}},{\text{ }} - 0.503} \right)\)     A1     N2

METHOD 2

attempt to use a coordinate to make an equation     (M1)

eg\(\,\,\,\,\,\)\(p\cos (6.25q) + r = 0.6,{\text{ }}p\cos (12.5q) + r = 1.5\)

correct substitution     (A1)

eg\(\,\,\,\,\,\)\(0.45\cos (6.25q) + 1.05 = 0.6,{\text{ }}0.45\cos (12.5q) + 1.05 = 1.5\)

0.502654

\(q = \frac{{4\pi }}{{25}},{\text{ }}0.503{\text{ }}\left( {{\text{or }} - \frac{{4\pi }}{{25}},{\text{ }} - 0.503} \right)\)     A1     N2

[3 marks]

b.ii.

valid method to find \(r\)     (M1)

eg\(\,\,\,\,\,\)\(\frac{{{\text{max}} + {\text{min}}}}{2},{\text{ }}0.6 + 0.45\)

\(r = 1.05\)     A1     N2

[2 marks]

b.iii.

METHOD 1

attempt to find start or end \(t\)-values for 12 December     (M1)

eg\(\,\,\,\,\,\)\(3 + 24,{\text{ }}t = 27,{\text{ }}t = 51\)

finds \(t\)-value for second max     (A1)

\(t = 50\)

23:00 (or 11 pm)     A1     N3

METHOD 2 

valid approach to list either the times of high tides after 21:00 or the \(t\)-values of high tides after 21:00, showing at least two times     (M1) 

eg\(\,\,\,\,\,\)\({\text{21:00}} + 12.5,{\text{ 21:00}} + 25,{\text{ }}12.5 + 12.5,{\text{ }}25 + 12.5\)

correct time of first high tide on 12 December     (A1)

eg\(\,\,\,\,\,\)10:30 (or 10:30 am) 

time of second high tide = 23:00     A1     N3

METHOD 3

attempt to set their \(h\) equal to 1.5     (M1)

eg\(\,\,\,\,\,\)\(h(t) = 1.5,{\text{ }}0.45\cos \left( {\frac{{4\pi }}{{25}}t} \right) + 1.05 = 1.5\)

correct working to find second max     (A1)

eg\(\,\,\,\,\,\)\(0.503t = 8\pi ,{\text{ }}t = 50\)

23:00 (or 11 pm)     A1     N3

[3 marks]

c.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
b.iii.
[N/A]
c.

Syllabus sections

Topic 3 - Circular functions and trigonometry » 3.5 » Solving trigonometric equations in a finite interval, both graphically and analytically.
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