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:
the radius of the base of the bucket is 9 cm,
the volume of the bucket is 3234π cm3.
[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
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)
\(\dfrac{(22\,309 \text{ to } 22\,324.3) \times 1.75}{8 \times 1000}\)
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)
We'd like to use Google Analytics cookies to count visits and see which pages help most. They're optional, and the site works exactly the same either way. Privacy policy
