(a) Calculate the volume of a sphere with radius 6 cm.
[The volume \(V\) of a sphere with radius \(r\) is \(V = \frac{4}{3}\pi r^3\).] [2]
(b) An ornament is made from a solid glass square-based pyramid. The base has side length 15 cm. A hemisphere with radius 6 cm is cut out of the base of the pyramid. This reduces the volume of glass contained in the ornament by 30%.
Calculate the perpendicular height of the pyramid.
[The volume of a pyramid is \(\frac{1}{3} \times\) area of base \(\times\) perpendicular height. A hemisphere is half a sphere.] [5]
Mark scheme (a)
Answer
Marks
Part marks and guidance
\(288\pi\) or 904.3 to 905
2
M1 for \(\frac{4}{3}\) (×) \(\pi\) (×) \(6^3\)
Accept 904 if M1 scored
Mark scheme (b)
Answer
Marks
Part marks and guidance
20.0[9...] to 20.1[...] or \(\dfrac{32}{5}\pi\) oe nfww
5
M1 for [hemisphere=] 0.5 × their (a) soi or 0.5 × \(\frac{4}{3}\) (×) \(\pi\) (×) \(6^3\) or [pyramid=] \(\frac{1}{3} \times 15 \times 15\) [×’\(h\)’] soi
Accept answer 20 after full working. No requirement at any stage for a formal equation. Values below provided as a guide to method being used, but mark method not accuracy: ie hemisphere (\(144\pi\) or 452.(..)) or pyramid (75[\(h\)])
M1 for [hemisphere=] 0.5 × their (a) soi or 0.5 × \(\frac{4}{3}\) (×) \(\pi\) (×) \(6^3\) and [pyramid=] \(\frac{1}{3} \times 15 \times 15\) [×’\(h\)’] soi OR 0.3 × their pyramid [×’\(h\)’] or \(\dfrac{\textit{their}\text{ hemisphere}}{0.3}\) oe
ie hemisphere (\(144\pi\) or 452.(..)) and pyramid (75[\(h\)])
OR 30% of pyramid (22.5[\(h\)]} or “reverse %” using hemisphere (\(480\pi\) or 1507(..))
M1 for hemisphere soi and 0.3 × their pyramid [×’\(h\)’]
OR \(\dfrac{\textit{their}\text{ hemisphere}}{0.3}\) oe and pyramid [×’\(h\)’] soi
ie hemisphere (\(144\pi\) or 452.(..)) and 30% of pyramid (22.5[\(h\)]} OR “reverse %” using hemisphere (\(480\pi\) or 1507(..)) and pyramid (75[\(h\)]). To receive M1M1M1 they should have both parts of the “ands” correct
M1 for \(\dfrac{\textit{their}\text{ hemisphere}}{0.3} \div\) their pyramid oe
If 0 scored, allow SC3 for \(\dfrac{64}{5}\pi\) or 40.[1..] to 40.2[…] as final answer
If correct, at this stage, it should be (\(480\pi\) or 1507(..)) ÷ 75 oe
