Surds: WJEC GCSE Numeracy Higher June 2023 Unit 1 Q7

Numeracy Higher June 2023 Unit 1 Q7

WJECHigherCurrent spec9 marksSurds Upper & Lower Bounds Volume & Surface Area

(a) Cartons of soft drink are filled using an automated system.

(i) Each carton is a cuboid that has:

a base area of exactly 40 cm²,
a height of 13 cm, correct to the nearest cm.

Calculate the least possible volume of a carton. [2]

(ii) The empty cartons, moving on a conveyor belt, pass under a filling gun that pours soft drink into the cartons until they are full.

Drink is poured into each carton at a rate of 8000 cm³ per minute, correct to the nearest 1000 cm³ per minute.

A filling gun above a conveyor belt carrying a row of cartons, the first four shaded

Diagram not drawn to scale

Calculate the greatest possible number of cartons that the automated system could fill in an hour.
You can ignore the time taken in moving from one full carton to the next empty one. [3]

(b) Cartons of a different size are also filled using the same system.
Each of these cartons has:

a square base,
a height of exactly 20 cm,
a volume of exactly 960 cm³.

Calculate the length of the side of the base of one of these cartons.
Give your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible. [4]

Mark scheme (a)

(i)

Scheme	Marks	Comments
(Least possible volume of a carton =) 40 × 12.5 = 500 (cm³)	M1 A1	Allow use of 12 < h < 13 for M1 only CAO If no marks awarded, SC1 for use of 12.5

(ii)

Scheme	Marks	Comments
Use of 8500	B1	Allow 8499.999(…) for 8500
\(\dfrac{8500}{500}\) (× 60)	M1	FT ‘their 500’ from (a)(i) If B0, FT provided unambiguously chosen: 8000 < ‘their 8500’ < 9000
= 1020	A1	CAO for ‘their 500’ from (a)(i)

Mark scheme (b)

Scheme	Marks	Comments
(length² =) 960 ÷ 20 length² = 48 OR (length =) √48	M1 A1	Or equivalent e.g. \(\sqrt{4} \times \sqrt{12}\)
(length =) 4√3	B2	Mark final answer Accept 40√3 mm, but not 0.04√3 m For B2, FT ‘their derived 48’ provided of equivalent difficulty (it has a square number as one of its factors) For B1, FT 'their derived 48' B1 for writing 48 as a product of 2 or more factors where one of the factors OR the product of 2 of their factors is a square number e.g. 16 × 3 or 4 × 4 × 3 or 4 × 12 or 2 × 2 × 12 OR B1 for writing \(\sqrt{48}\) as a product of 2 or more factors where one of the factors OR the product of 2 of their factors gives a whole number e.g. \(\sqrt{16} \times \sqrt{3}\) or \(\sqrt{2} \times \sqrt{2} \times \sqrt{12}\) or \(2\sqrt{12}\)