(a) Dee-Odd makes cocktail glasses out of thin glass. One of their cocktail glasses is shown below. A hollow hemisphere forms the part of the glass which can hold the drink.
Diagram not drawn to scale
Before each hemisphere is attached to a stem, the surface area on the outside of each hemisphere is given a decorative coating.
The volume of each hemisphere is \(\dfrac{128\pi}{3}\) cm3.
Calculate the surface area that is given a decorative coating. Give your answer in terms of π in its simplest form. [6]
(b) Dee-Odd also makes another style of cocktail glass in two different sizes. These glasses are mathematically similar.
Diagram not drawn to scale
The ratio of the volume of the smaller glass to the volume of the larger glass is 8 : 27. The height of the larger glass is 18 cm. Calculate the height of the smaller glass. [3]
From the formula list: Volume of sphere \(= \dfrac{4}{3}\pi r^3\); Surface area of sphere \(= 4\pi r^2\)
Mark scheme (a)
Scheme
Marks
Comments
\(\dfrac{2}{3}\) × π × radius3 = \(\dfrac{128\pi}{3}\) or equivalent
M1
radius3 = \(\dfrac{128\pi \times 3}{2 \times \pi \times 3}\) or equivalent OR radius = \(\sqrt[3]{\dfrac{128\pi \times 3}{2 \times \pi \times 3}}\) or equivalent
M1
For isolating radius3 FT if the volume of a sphere used for a possible M1A1 only
(radius =) 4 (cm)
A2
CAO A1 for radius3 = 64 or (radius =) \(\sqrt[3]{64}\) OR A1 for radius3 = 32 or (radius =) \(\sqrt[3]{32}\) if the volume of a sphere used
(Surface area =) 2 × π × 42
M1
FT ‘their 4’ provided an attempt made to use the formula for the volume of a sphere to calculate the radius
= 32π (cm2)
A1
Accept 3200π mm2 or 0.0032π m2 If ‘their radius = \(\sqrt[3]{32}\)’ then their SA will be 213/3 π
If final M0A0, award SC1 for 4 × π × ‘their r2’ evaluated correctly provided an attempt made to use the formula for the volume of a sphere to calculate the radius
Mark scheme (b)
Scheme
Marks
Comments
\(\sqrt[3]{8} : \sqrt[3]{27}\) or \(\sqrt[3]{8 \div 27}\) or \(\sqrt[3]{27 \div 8}\) ( = 2 : 3) (= 2/3) (= 3/2)
B1
May be implied in further working
(Height =) 18 × \(\dfrac{2}{3}\) or 18 ÷ \(\dfrac{3}{2}\) or equivalent
M1
Implies previous B1 provided not from incorrect working FT ‘their 2/3’ or ‘their 3/2’ provided first B1 awarded
= 12 (cm)
A1
CAO
Note: 27 ÷ 18 = 1.5 followed by either 18 ÷ 1.5 = 12 or 8 × 1.5 = 12 is awarded B0M0A0
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