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Date November 2021 Marks available 3 Reference code 21N.2.SL.TZ0.8
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Determine Question number 8 Adapted from N/A

Question

The height of water, in metres, in Dungeness harbour is modelled by the function H(t)=asin(b(t-c))+d, where t is the number of hours after midnight, and a, b, c and d are constants, where a>0, b>0 and c>0.

The following graph shows the height of the water for 13 hours, starting at midnight.

The first high tide occurs at 04:30 and the next high tide occurs 12 hours later. Throughout the day, the height of the water fluctuates between 2.2m and 6.8m.

All heights are given correct to one decimal place.

Show that b=π6.

[1]
a.

Find the value of a.

[2]
b.

Find the value of d.

[2]
c.

Find the smallest possible value of c.

[3]
d.

Find the height of the water at 12:00.

[2]
e.

Determine the number of hours, over a 24-hour period, for which the tide is higher than 5 metres.

[3]
f.

Markscheme

12=2πb  OR  b=2π12                A1

b=π6                AG

 

[1 mark]

a.

a=6.8-2.22  OR  a=max-min2                (M1)

=2.3m                A1

 

[2 marks]

b.

d=6.8+2.22  OR  d=max+min2                (M1)

=4.5m                A1

 

[2 marks]

c.

METHOD 1

substituting t=4.5 and H=6.8 for example into their equation for H                (A1)

6.8=2.3sinπ64.5-c+4.5

attempt to solve their equation                (M1)

c=1.5                A1

 

METHOD 2

using horizontal translation of 124                (M1)

4.5-c=3                (A1)

c=1.5                A1

 

METHOD 3

H't=2.3π6cosπ6t-c                (A1)

attempts to solve their H'4.5=0 for c                (M1)

2.3π6cosπ64.5-c=0

c=1.5                A1

 

[3 marks]

d.

attempt to find H when t=12 or t=0, graphically or algebraically                (M1)

H=2.87365

H=2.87m                A1

 

[2 marks]

e.

attempt to solve 5=2.3sinπ6t-1.5+4.5                (M1)

times are t=1.91852 and t=7.08147 , t=13.9185, t=19.0814                (A1)

total time is 2×7.081-1.919

10.3258

=10.3 (hours)                A1


Note: Accept 10.

 

[3 marks]

f.

Examiners report

[N/A]
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b.
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c.
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d.
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e.
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f.

Syllabus sections

Topic 2—Functions » SL 2.4—Key features of graphs, intersections using technology
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Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations
Topic 2—Functions
Topic 3— Geometry and trigonometry

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