Circles & Sectors: WJEC GCSE Numeracy Higher June 2023 Unit 2 Q8

Numeracy Higher June 2023 Unit 2 Q8

WJECHigherCurrent spec6 marksCircles & Sectors

8. Central Pivot Irrigation is a method of watering crops.
The sprinkler arm, supported by wheeled towers, rotates around a central pivot.
For each complete rotation, the watered region forms a circle as shown below.

Circle with the Pivot at its centre; the Sprinkler arm is a radius with towers marked along it, and the Outer tower is on the circumference

Diagram not drawn to scale

A typical Central Pivot Irrigation system waters a circular region of radius 400 m.
The sprinkler arm rotates through an angle of 30° per hour.

(a)

Calculate the area of the region that is watered in one hour. [2]

(b)

The outer tower is 400 m from the pivot.
During one watering session, the outer tower moves 1067·6 m.
Calculate the angle the sprinkler arm has turned through. [3]

(c)

The part of the sprinkler arm furthest away from the pivot delivers water at a faster rate than the part of the sprinkler arm closest to the pivot.

Give one reason why the system has been designed in this way. [1]

Scheme	Marks	Comments
\(\dfrac{30}{360} \times \boldsymbol{\pi} \times 400^2\) or equivalent	M1
= 41 866 to 41 893.4 (m\(^{2}\))	A1	Or \(\dfrac{40000\boldsymbol{\pi}}{3}\)

Mark scheme (b)

Scheme	Marks	Comments
\(\dfrac{\text{Angle}}{360} \times 2 \times \boldsymbol{\pi} \times 400 = 1067.6\)	M1	May be implied in further work
(Angle =) \(\dfrac{1067.6 \times 360}{2 \times \boldsymbol{\pi} \times 400}\) or equivalent	m1	For isolating the angle
= 152.9 to 153 (º)	A1	If no marks, award SC1 for an answer of 305.8 to 306 (º) from using the calculation \(\dfrac{1067.6 \times 360}{\boldsymbol{\pi} \times 400}\)

Mark scheme (c)

Scheme	Marks	Comments
Valid reason e.g. ‘More area to water the further away from the pivot you are’, ‘The outer tower will be moving faster’ ‘It has to cover a larger distance’, ‘The part closest to the pivot needs to deliver less water or it would flood’	E1	Ignore spurious additional comments Accept e.g. ‘To cover more area’ Allow e.g. ‘So all the crops get the same amount of water’ Do not accept e.g. ‘The crops furthest away need more water’, ‘To keep up with the tower closest to the pivot as this doesn’t move so far’, ‘So the water will spray at a constant speed’