Similar Shapes: WJEC GCSE Numeracy Higher June 2019 Unit 2 Q8

Numeracy Higher June 2019 Unit 2 Q8

8. A company makes buckets in two sizes. Both sizes are in the shape of a frustum of a cone.
Bucket A has the dimensions shown in the diagram below.

Diagram not drawn to scale

From the formula list: Volume of cone \(= \dfrac{1}{3}\pi r^2 h\)

(a) Show that:

[5]

(b) Bucket B is shown below. It is mathematically similar to Bucket A.

Diagram not drawn to scale

Calculate the number of gallons Bucket B can hold when full. [6]

Remember:
1 gallon = 8 pints

Mark scheme (a)

Scheme	Marks	Comments
(radius =) 15 × 33 ÷ (22 + 33) or 15 × ³/₅ (= 9) or equivalent	B2	Working MUST be shown here May be seen with appropriate tangent ratios If Pythagoras used, appropriate use of the scale factor would be needed Allow B1 for sight of ³³/₅₅ or equivalent OR ⁵⁵/₃₃ or equivalent
Sight of 1/3×π×15²×55 OR 1/3×π×9²×33	B1	(12952 to 12961 OR 2797.7 to 2800)
1/3×π×15²×55 – 1/3×π×9²×33 = 4125π – 891π (= 3234π (cm³))	M1 A1	Accept values in the range 10154.7 to 10161.2 (cm³)

Mark scheme (b)

Scheme	Marks	Comments
(Scale factor =) \(\dfrac{\text{28.6}}{\text{22}}\) (=1.3) OR \(\dfrac{\text{22}}{\text{28.6}}\) (=0.769…)	B1
(Volume factor =) (28.6/22)³ or 1.3³ OR (22/28.6)³ or 0.769…³	B1	Or 2.197 Or 0.455…
3234π × (28.6/22)³ or ÷ (22/28.6)³ OR \(\dfrac{3234\pi \times 1.75}{8 \times 1000}\)	M1	Accept numerical values for 3234π (10154.7 to 10161.2) Allow use of the conversion 1 litre = 1.75 to 1.76 pints
= 22309 to 22324.3 OR = 2.22 to 2.26 (or 7105(.098)π to 7105.1π) (or 0.707π to 0.719π)	A1
\(\dfrac{\times 1.75}{8 \times 1000}\) OR × (28.6/22)³ or ÷ (22/28.6)³	M1	Allow use of the conversion 1 litre = 1.75 to 1.76 pints FT ‘their derived 22309 to 22324.3’ OR FT ‘their 2.2(2…)’ from use of 3234π
= 4.8(8…) to 4.96(…) (gallons)	A1	Allow an answer of 5 (gallons) from correct working Allow 1.55π to 1.58π (gallons). Do not accept 1.6π Allow the conversion into gallons for the M1 mark from any of the following also 1 pint = 567 to 570 ml: ÷ 8 ÷ (567 to 570) 1 gallon = 4.5 to 4.55 litres: ÷ 1000 ÷ (4.5 to 4.55) 1 litre = 0.219 to 0.22 gallons: ÷ 1000 × (0.219 to 0.22)

Alternative method:

Scheme	Marks	Comments
(Scale factor =) \(\dfrac{\text{28.6}}{\text{22}}\) (=1.3)	B1
Dimensions of 19.5, 71.5, 11.7, 42.9	B1	Allow B1 for any 3 correct dimensions
1/3×π×19.5²×71.5 – 1/3×π×11.7²×42.9	M1	(28456 to 28475) – (6146 to 6151)
= 22 309 to 22 324.3 (cm³)	A1	Or 7105(.098)π to 7105.1π
\(\dfrac{(22\,309 \text{ to } 22\,324.3) \times 1.75}{8 \times 1000}\)	M1	Allow use of the conversion 1 litre = 1.75 to 1.76 pints FT ‘their derived 22309 to 22324.3’
= 4.8(8…) to 4.96(…) (gallons)	A1	Allow an answer of 5 (gallons) from correct working Allow 1.55π to 1.58π (gallons). Do not accept 1.6π Allow the conversion into gallons for the M1 mark from any of the following also 1 pint = 567 to 570 ml: ÷ 8 ÷ (567 to 570) 1 gallon = 4.5 to 4.55 litres: ÷ 1000 ÷ (4.5 to 4.55) 1 litre = 0.219 to 0.22 gallons: ÷ 1000 × (0.219 to 0.22)