Higher June 2023 Paper 1 Q17
17 A solid hemisphere has radius \(x\).
A solid cylinder has radius \(3x\) and height \(x\).
| Surface area of a sphere \(= 4\pi r^2\) where \(r\) is the radius |
Work out the ratio
total surface area of the hemisphere : total surface area of the cylinder
Give your answer in its simplest form.
You must show your working. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – expressions in \(x\) | ||
| \(4\pi x^2 \div 2\) or \(2\pi x^2\) or \(\pi x^2\) or \(\pi(3x)^2\) or \(9\pi x^2\) or \(2 \times \pi(3x)^2\) or \(18\pi x^2\) or \(2\pi x(3x)\) or \(6\pi x^2\) | M1 | oe area of curved face of hemisphere oe area of flat face of hemisphere oe area of one flat face of cylinder oe area of both flat faces of cylinder oe area of curved face of cylinder |
| \(4\pi x^2 \div 2 + \pi x^2\) or \(3\pi x^2\) or \(\pi(3x)^2 + \pi(3x)^2 + 2\pi x(3x)\) or \(9\pi x^2 + 9\pi x^2 + 6\pi x^2\) or \(24\pi x^2\) | M1dep | oe total surface area of the hemisphere oe total surface area of the cylinder |
| \(3\pi x^2\) and \(24\pi x^2\) and 1 : 8 | A1 | either order |
| Alternative method 2 – substituting a value for \(x\) | ||
| Substitutes a value for \(x\) and works out the area of at least one of area of curved face of hemisphere area of flat face of hemisphere area of one flat face of cylinder area of both flat faces of cylinder area of curved face of cylinder | M1 | eg using \(x = 5\), at least one of \(50\pi\) \(25\pi\) \(225\pi\) \(450\pi\) \(150\pi\) |
| Substitutes a value for \(x\) and works out an expression for the total surface area of the hemisphere or the cylinder | M1dep | eg using \(x = 5\) total surface area of hemisphere \(= 25\pi + 50\pi\) or \(75\pi\) or total surface area of cylinder \(= 225\pi + 225\pi + 150\pi\) or \(600\pi\) |
| Both correct total surface areas for their value of \(x\) and 1 : 8 | A1 | either order |
Additional guidance
| 1 : 8 or 8 : 1 without correct working or values | M0M0A0 |
| Condone \(\pi\) missing consistently for all marks | |
| Allow ‘correct’ and consistent values of \(\pi\) throughout (eg 3, 3.14, \(\dfrac{22}{7}\)) | |
| Condone use of \(r\) for \(x\) throughout | |
| Do not allow \(3\pi x^2\) from \(3x \times \pi \times x\) oe |