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Date	November 2020	Marks available	3	Reference code	20N.2.SL.TZ0.S_8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	S_8	Adapted from	N/A

Question

The following diagram shows a water wheel with centre $O$ and radius $10$ metres. Water flows into buckets, turning the wheel clockwise at a constant speed.

The height, $h$ metres, of the top of a bucket above the ground $t$ seconds after it passes through point $A$ is modelled by the function

$h (t) = 13 + 8 \cos (\frac{π}{18} t) - 6 \sin (\frac{π}{18} t)$ , for $t \geq 0$ .

A bucket moves around to point $B$ which is at a height of $4.06$ metres above the ground. It takes $k$ seconds for the top of this bucket to go from point $A$ to point $B$ .

The chord $[AB]$ is $17.0$ metres, correct to three significant figures.

Find the height of point $A$ above the ground.

[2]

a.i.

Calculate the number of seconds it takes for the water wheel to complete one rotation.

[2]

a.ii.

Hence find the number of rotations the water wheel makes in one hour.

[2]

a.iii.

Find $k$ .

[3]

Find $A \hat{O} B$ .

[3]

Determine the rate of change of $h$ when the top of the bucket is at $B$ .

[2]

Markscheme

valid approach (M1)

eg $h (0), 13 + 8 \cos (\frac{π}{18} \times 0) - 6 \sin (\frac{π}{18} \times 0), 13 + 8 \times 1 - 6 \times 0$

$21$ (metres) A1 N2

[2 marks]

a.i.

valid approach to find the period (seen anywhere) (M1)

eg $(36, 21)$ , attempt to find two consecutive max/min, $50.3130 - 14.3130$

$\frac{2 π}{\frac{π}{18}}, b = \frac{2 π}{period},$

$36$ (seconds) (exact) A1 N2

[2 marks]

a.ii.

correct approach (A1)

eg $\frac{60 \times 60}{36}, 1.6666$ rotations per minute

$100$ (rotations) A1 N2

[2 marks]

a.iii.

correct substitution into equation (accept the use of $t$ ) (A1)

eg $4.06 = 13 + 8 \cos (\frac{π}{18} \times k) - 6 \sin (\frac{π}{18} \times k)$

valid attempt to solve their equation (M1)

$11.6510$

$11.7$ A1 N3

[3 marks]

METHOD 1

evidence of choosing the cosine rule or sine rule (M1)

eg ${AB}^{2} = {OA}^{2} + {OB}^{2} - 2 \times OA \times OB \cos (A \hat{O} B), \frac{\sin (A \hat{O} B)}{AB} = \frac{\sin (O \hat{A} B)}{OB}$

correct working (A1)

eg $\cos (A \hat{O} B) = \frac{10^{2} + 10^{2} - 17 . 0^{2}}{2 \times 10 \times 10}, - 0.445, \frac{\sin (A \hat{O} B)}{17.0} = \frac{\sin (\frac{π}{2} - \frac{1}{2} A \hat{O} B)}{10},$

$\frac{\sin (O \hat{A} B)}{10} = \frac{\sin (π - 2 \times O \hat{A} B)}{17.0}$

$2.03197, 116.423 °$

$2.03 (116 °)$ A1 N3

METHOD 2

attempt to find the half central angle (M1)

eg $\sin (\frac{1}{2} A \hat{O} B) = \frac{\frac{1}{2} AB}{OA}$

correct working (A1)

eg $2 \times \sin^{- 1} (\frac{8.5}{10})$

$2.03197, 116.423 °$

$2.03 (116 °)$ A1 N3

METHOD 3

valid approach to find fraction of period (M1)

eg $\frac{k}{36}, \frac{11.6510}{36}$

correct approach to find angle (A1)

eg $\frac{k}{36} \times 2 π$

$2.03348, 116.510 °$ ( $2.04203$ using $11.7$ )

$2.03 (117 °)$ A1 N3

[3 marks]

recognizing rate of change is $h'$ (M1)

eg $h' (k), h' (11.6510), 0.782024$

$- 0.782024$ ( $- 0.768662$ from $3 sf$ )

rate of change is $- 0.782 ({ms}^{- 1})$ A1 N2

( $- 0.769 ({ms}^{- 1})$ from $3 sf$ )

[2 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

a.iii.

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations

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22M.2.SL.TZ2.8b:
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EXM.3.AHL.TZ0.5a.iii:
Write down the period of this graph
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18M.2.SL.TZ1.S_10b.i:
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17M.2.SL.TZ2.S_4c:
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17N.2.SL.TZ0.S_10a:
Show that $f(2\pi ) = 2\pi$ .
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Show that $\frac{a}{b} = {\text{tan}}\,\theta$ .
17M.2.SL.TZ1.S_8a.i:
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EXM.3.AHL.TZ0.5c.ii:
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EXM.3.AHL.TZ0.5e.i:
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EXM.3.AHL.TZ0.5b.ii:
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17N.2.SL.TZ0.S_10d:
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EXM.3.AHL.TZ0.5e.iii:
$C$ .
18M.2.SL.TZ1.S_10e:
Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².
19M.1.AHL.TZ2.H_9a:
Find the $x$ -coordinates of the points of intersection of the two graphs.
19M.2.AHL.TZ1.H_10d:
Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left( t \right)$ ≥ 3.
17M.2.SL.TZ1.S_8a.ii:
Find the difference in height between low tide and high tide.
17N.2.SL.TZ0.S_10b.ii:
Find the equation of $L$ .
18M.2.SL.TZ1.S_10d:
Find the maximum speed of the ball.
19M.2.AHL.TZ1.H_10g:
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EXM.3.AHL.TZ0.5a.i:
Sketch the graph $y = 3\,{\text{sin}}\,x + 4\,{\text{cos}}\,x$ , for $- 2\pi \leqslant x \leqslant 2\pi$
16N.2.SL.TZ0.S_10a:
(i) Find the value of $c$ .

(ii) Show that $b = \frac{\pi }{6}$ .

(iii) Find the value of $a$ .
19M.1.SL.TZ2.S_7b:
Consider the graph of $g\left( x \right) = 2\,{\text{sin}}\,px$ , 0 ≤ $x$ < $2\pi$ , where $p$ > 0.

Find the greatest value of $p$ such that the graph of $g$ does not intersect the line $y = - 1$ .
EXM.3.AHL.TZ0.5c.i:
Find ${\text{arctan}}\frac{3}{4}$ , giving the answer to 3 significant figures.
EXM.3.AHL.TZ0.5g:
Hence prove your conjectures in part (e).
17N.2.SL.TZ0.S_10b.i:
Find the coordinates of ${{\text{P}}_0}$ and of ${{\text{P}}_1}$ .
22M.1.SL.TZ1.6b:
The $y$ -intercept of the graph of $g$ is at $(0, r)$ .

Given that $g (x) \geq 7$ , find the smallest value of $r$ .
EXN.1.SL.TZ0.9b:
Find the minimum height of the ball above the ground.
EXN.1.SL.TZ0.9d:
For the first 2 seconds of its motion, determine the amount of time that the ball is less than $1.8 + 0.2 \sqrt{2}$ metres above the ground.
17N.2.SL.TZ0.S_10c:
Show that the distance between the $x$ -coordinates of ${{\text{P}}_k}$ and ${{\text{P}}_{k + 1}}$ is $2\pi$ .
19M.2.AHL.TZ1.H_10b:
Write down two transformations that will transform the graph of $y = v\left( t \right)$ onto the graph of $y = i\left( t \right)$ .
22M.2.AHL.TZ2.10b:
Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .
20N.2.SL.TZ0.S_8a.iii:
Hence find the number of rotations the water wheel makes in one hour.
EXM.3.AHL.TZ0.5b.i:
Use your answers from part (a) to write down the value of $A$ , $B$ and $D$ .
20N.2.SL.TZ0.S_8d:
Determine the rate of change of $h$ when the top of the bucket is at $B$ .
EXM.3.AHL.TZ0.5f.i:
Show that $\frac{b}{{\sqrt {{a^2} + {b^2}} }} = {\text{cos}}\,\theta$ .
19M.2.SL.TZ1.S_8c.ii:
Find the period of $g$ .
19M.2.SL.TZ1.S_8d:
The equation $g\left( x \right) = 12$ has two solutions where $\pi$ ≤ $x$ ≤ ${\frac{{4\pi }}{3}}$ . Find both solutions.
18M.2.SL.TZ1.S_10b.ii:
For the graph of $f$ , write down the period.
19M.1.AHL.TZ2.H_9c:
At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the $y$ -coordinate of A on the graph of $g$ .
17M.2.SL.TZ2.S_4a:
Find the value of $p$ .
SPM.2.SL.TZ0.9b.ii:
Hence, find the value of $f$ (6).
21M.1.SL.TZ2.8a:
Find the $x$ -coordinates of $A$ and $B$ .
19M.1.AHL.TZ2.H_9b:
Find the exact area of the shaded region, giving your answer in the form $p\pi + q\sqrt 3$ , where $p$ , $q \in \mathbb{Q}$ .
17M.2.SL.TZ1.S_8b.ii:
Find the value of $q$ ;
SPM.2.SL.TZ0.9b.i:
Find the value of $b$ .
EXN.1.SL.TZ0.9a:
Find the height of the ball above the ground when it is released.
EXN.1.SL.TZ0.9c:
Show that the ball takes $2$ seconds to return to its initial height above the ground for the first time.
20N.2.SL.TZ0.S_8c:
Find $A \hat{O} B$ .
19M.2.AHL.TZ1.H_10e:
Find ${p_{av}}$ (0.007).
20N.2.SL.TZ0.S_8a.ii:
Calculate the number of seconds it takes for the water wheel to complete one rotation.
20N.2.SL.TZ0.S_8a.i:
Find the height of point $A$ above the ground.
19M.2.AHL.TZ1.H_10f:
With reference to your graph of $y = p\left( t \right)$ explain why ${p_{av}}\left( T \right)$ > 0 for all $T$ > 0.
17M.2.SL.TZ2.S_4b:
Find the value of $q$ .
19M.2.SL.TZ1.S_8a:
The range of $f$ is $k$ ≤ $f\left( x \right)$ ≤ $m$ . Find $k$ and $m$ .
21M.2.SL.TZ2.7d:
Find the value of $p$ and the value of $q$ .
16N.2.SL.TZ0.S_10c:
(i) Find $w$ .

(ii) Hence or otherwise, find the maximum positive rate of change of $g$ .
21M.2.SL.TZ2.7e:
If the observer stands directly under point $C$ for one minute, point $D$ will pass over his head $n$ times.

Find the value of $n$ .
21N.2.SL.TZ0.8a:
Show that $b = \frac{π}{6}$ .
21N.2.SL.TZ0.8c:
Find the value of $d$ .
21N.2.SL.TZ0.8d:
Find the smallest possible value of $c$ .
22M.1.SL.TZ1.6a:
Describe these two transformations.
21N.2.SL.TZ0.8b:
Find the value of $a$ .
21N.2.AHL.TZ0.9a:
Show that $b = \frac{π}{6}$ .
21N.2.AHL.TZ0.9b:
Find the value of $a$ .
21N.2.AHL.TZ0.9d:
Find the smallest possible value of $c$ .
EXM.3.AHL.TZ0.5a.ii:
Write down the amplitude of this graph
EXM.3.AHL.TZ0.5d:
By considering the graph of $y = 5\,{\text{sin}}\,x + 12\,{\text{cos}}\,x$ , find the value of $A$ , $B$ , $C$ and $D$ .
18M.2.SL.TZ1.S_10c:
Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .
17M.2.SL.TZ1.S_8b.i:
Find the value of $p$ ;
EXM.3.AHL.TZ0.5e.ii:
$B$ .
EXM.3.AHL.TZ0.5e.iv:
$D$ .
17M.2.SL.TZ1.S_8b.iii:
Find the value of $r$ .
17M.2.SL.TZ1.S_8c:
There are two high tides on 12 December 2017. At what time does the second high tide occur?
16N.2.SL.TZ0.S_10b:
(i) Write down the value of $k$ .

(ii) Find $g(x)$ .
SPM.2.SL.TZ0.9d:
Find the value of $x$ for which the functions have the greatest difference.
SPM.2.SL.TZ0.9c:
Find the value of $p$ and the value of $q$ .
21M.2.SL.TZ1.4:
A Ferris wheel with diameter $110$ metres rotates at a constant speed. The lowest point on the wheel is $10$ metres above the ground, as shown on the following diagram. $P$ is a point on the wheel. The wheel starts moving with $P$ at the lowest point and completes one revolution in $20$ minutes.

The height, $h$ metres, of $P$ above the ground after $t$ minutes is given by $h (t) = a \cos (b t) + c$ , where $a, b, c \in ℝ$ .

Find the values of $a$ , $b$ and $c$ .
21N.2.AHL.TZ0.9e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9c:
Find the value of $d$ .
21N.2.AHL.TZ0.9f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9g:
A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.

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Topic 3— Geometry and trigonometry

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