Basic Trigonometry: WJEC GCSE Numeracy Higher June 2023 Unit 2 Q5

Numeracy Higher June 2023 Unit 2 Q5

WJECHigherCurrent spec7 marksBasic Trigonometry Volume & Surface Area

5. The cost of sending a package depends on its volume.

Volume (cm\(^3\))	Cost
0 to 1000	£12.55
greater than 1000, up to 2000	£13.60
greater than 2000, up to 4000	£14.85
greater than 4000, up to 10 000	£16.25
Parcels with volume greater than 10 000 cm\(^3\) are not accepted

A rectangular ‘HANDLE WITH CARE’ label is stuck on a package.
The label measures 17·5 cm by 11·1 cm.
The package is a cuboid with height 19 cm and width 6·7 cm.
Each vertex of the label touches an edge of the front face of the package.

Cuboid package with depth 6·7 cm and height 19 cm; the horizontal front edge is labelled Length. A tilted rectangular label HANDLE WITH CARE, 17·5 cm by 11·1 cm, has one vertex on the bottom edge, where the 11·1 cm side makes 56° and the 17·5 cm side makes 34° with the bottom edge; its other vertices touch the left, top and right edges of the front face

Diagram not drawn to scale

Calculate the volume of the package and hence find the cost of sending this package.
You must show all your working. [7]

The volume of the package is ……………………………
The cost of sending this package is ……………………………

Mark scheme

Scheme

Marks

Comments

(Length of the package, x + y)
(x =) 17.5 × cos34° or (x =) 17.5 × sin56°
AND
(y =) 11.1 × cos56° or (y =) 11.1. × sin34°

Or alternative full method
M2 for any 1 of these statements correct or as appropriate from an alternative method
OR
M1 for .../17.5 = cos34° or .../17.5 = sin56°,
or .../11.1 = cos56° or .../11.1 = sin34°

Sight of 14.5(08... cm) and 6.2(07.... cm) or
for the sum of these: 20.7(… cm) or 21 (cm)

Must be from correct working (not from 11.1\(^{2}\) + 17.5\(^{2}\))
A1 for 14.5(08... cm) or 6.2(07.... cm)

(Volume =) 19 × 6.7 × (14.5(08...) + 6.2(07....))
or 19 × 6.7 × 20.7
or 19 × 6.7 × 21

FT ‘their x + y’ provided some use of trigonometry attempted previously (including incorrect use) and both x > 0 and y > 0.
Award M1 for an unsupported correct volume, or ‘their FT volume’ provided FT criteria met

Answer in the range 2635 (cm\(^{3}\)) to 2673.5 (cm\(^{3}\))
AND Cost (£)14.85

Answer space takes precedence
FT from truncation or rounding
FT for appropriate cost for ‘their volume’ provided it is ≤10 000 (cm\(^{3}\))
FT is 127.3 × ‘their x + their y’ correctly evaluated

Volume (cm\(^{3}\))	Cost
0 to 1000	£12.55
greater than 1000, up to 2000	£13.60
greater than 2000, up to 4000	£14.85
greater than 4000, up to 10 000	£16.25

If ‘y’ not considered, possible M2, A1 then also award SC1 for a volume of 1845 (cm\(^{3}\)) to 1847.2 (cm\(^{3}\)) AND cost (£)13.6(0)

If ‘x’ not considered, possible M2, A1 then also award SC1 for a volume of 789 (cm\(^{3}\)) to 790.6 (cm\(^{3}\)) AND cost (£)12.55

If no marks, award SC1 for an answer in the range 2635 (cm\(^{3}\)) to 2673.5 (cm\(^{3}\)) AND Cost (£)14.85 from use of 20.7…(cm) from \(\sqrt{(11.1^2 + 17.5^2)}\)