Circles & Sectors: WJEC GCSE Numeracy Higher November 2024 Unit 1 Q9

Numeracy Higher November 2024 Unit 1 Q9

WJECHigherCurrent spec13 marksCircles & Sectors Recurring Decimals Similar Shapes

9. Hang-Up is a company that makes clothes hangers.
One type of hanger is made from a single length of metal wire that is bent to form the hanger.

(a)

(i) The design for the main part of the hanger is shown below.
This part of the hanger is symmetrical.
It has straight sections of length 24 cm and 38 cm.
It also has curved sections that are arcs of a circle of radius 3 cm.

Hanger main part: two sloping straight sections of 24 cm meeting at the top, a straight base of 38 cm, and a curved arc at each end of radius 3 cm with angle 150° at the centre

Diagram not drawn to scale

Calculate the length of metal wire needed for the main part of the hanger.
Give your answer in terms of \(\pi\) in its simplest form. [4]

(ii) The length of wire needed for the hooked part of this hanger is \(0{\cdot}1\dot{3}\) of the total length of wire needed for a hanger.

Write \(0{\cdot}1\dot{3}\) as a fraction in its simplest form. [3]

(b) Hang-Up makes metal hangers in batches of 80.
Every 3 hours, it randomly samples 8 hangers from one batch of 80 for quality assurance.

The following numbers are taken from a table of random digits.

299986 890791 810130 955579 268884 301244

Use these numbers to choose 8 hangers from a batch of 80 hangers.
You must start with the first number in the list.
Describe clearly how you use the numbers to select the sample. [3]

Hangers chosen:

(c) Hang-Up also makes plastic hangers of various sizes.
Two of its plastic hangers have main parts that are mathematically similar.
These are shown below.

Two mathematically similar plastic hangers; the smaller one has height 9 cm

Diagram not drawn to scale

The total surface area of the larger hanger is 1·44 times the total surface area of the smaller hanger.

The height of the smaller hanger is 9 cm.
Calculate the height of the larger hanger. [3]

Mark scheme (a)

Scheme	Marks	Comments
(i) Sight of \(\dfrac{150{\times}2{\times}\boldsymbol{\pi}{\times}3}{360}\) OR \(\dfrac{300{\times}2{\times}\boldsymbol{\pi}{\times}3}{360}\) (= 2.5\(\boldsymbol{\pi}\)) (= 5\(\boldsymbol{\pi}\))	B1	Or equivalents
(Length of wire =) (2 ×) \(\dfrac{150{\times}2{\times}\boldsymbol{\pi}{\times}3}{360}\) + (2 ×) 24 + 38 or equivalent	M1
= 5\(\boldsymbol{\pi}\) + 86 (cm)	A2	CAO Mark final answer A1 for sight of any one of the following: \(\dfrac{1800\boldsymbol{\pi}}{360}\) + 86 or equivalent 5\(\boldsymbol{\pi}\) + …. 2.5\(\boldsymbol{\pi}\) + …. or 5\(\boldsymbol{\pi}\)/2 + …. provided an attempt has been made to add the 3 straight pieces
(ii) e.g. 10x = 1.333…, 100x = 13.333… AND an attempt to subtract both sides	M1
= \(\dfrac{12}{90}\) or \(\dfrac{132}{990}\) or \(\dfrac{1332}{9990}\) or equivalent	A1	Allow A1 for e.g. 1.2/9
= \(\dfrac{2}{15}\)	A1	FT from M1A0 provided of equivalent difficulty

Alternative method (a)(ii)

Scheme	Marks	Comments
\(\dfrac{1}{10}\) + \(\dfrac{3}{90}\) or equivalent	M1
= \(\dfrac{12}{90}\) or equivalent	A1
= \(\dfrac{2}{15}\)	A1	FT from M1A0 provided of equivalent difficulty

Mark scheme (b)

Scheme	Marks	Comments
Statements required: Number the hangers from (0)1 to 80 Consider successive 2-digit numbers Do not use numbers outside the range e.g. Do not use 00 and 81 – 99, OR Use the numbers (0)1 to 80 Ignore repeats	E2	All 4 needed for E2 E1 for any 2 or 3 correct statements Allow an equivalent numbering system e.g. (0)0 to 79 Their numbering system can be implied by the range of numbers they state they will choose from Allow the 2^nd statement to be implied by their numbering of the hangers (from 01) AND their use of 2-digit numbers in their answer OR 2-digit numbers used in their answer with 07 and 01 seen Do not allow ‘Use numbers less than 81’ if they have numbered the hangers from 01 to 80, without stating that 00 will not be used
(Clothes hangers chosen =) 29, (0)7, (0)1, 30, 55, 79, 26, 30, 12	B1	ISW Answer space takes precedence

Mark scheme (c)

Scheme	Marks	Comments
(Scale factor =) \(\sqrt{1.44}\) or 1.2	B1
(Height of larger hanger =) \(\sqrt{1.44}\) × 9 or 1.2 × 9	M1	FT ‘their \(\sqrt{1.44}\)’
= 10.8 (cm)	A1	CAO