Higher January 2019 Paper 1 Q15
15 The total surface area of a solid hemisphere is equal to the curved surface area of a cylinder.
The radius of the hemisphere is \(r\) cm.
The radius of the cylinder is twice the radius of the hemisphere.
Given that
volume of hemisphere : volume of cylinder = 1 : \(m\)
find the value of \(m\).
(4)
| Scheme | Marks |
|---|---|
| Eg \(\dfrac{4\pi r^2}{2}\,(+\pi r^2) = 2\pi(2r)h\) oe | M1 |
| \(h = \dfrac{3}{4}r\) or \(r = \dfrac{4}{3}h\) | A1 |
Eg \(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi \times r^3\) and \(\pi \times (2r)^2 \times \text{``}{\dfrac{3}{4}r}\text{''}\) OR \(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi \times \left(\text{``}{\dfrac{4}{3}h}\text{''}\right)^3\) and \(\pi \times \left(2 \times \text{``}{\dfrac{4}{3}h}\text{''}\right)^2 \times h\) | M1 |
| 4.5 oe | A1 |
| (4) | |
| (4 marks) |
Notes
M1: for use of, for example, \(r\) and \(2r\) in an equation condone omission of flat surface area
A1: for a correct expression for either \(r\) or \(h\)
M1: dep on award of first M1
ft for candidate’s expression for \(r\) or \(h\) for correct expressions for volume of hemisphere and volume of cylinder; both in terms of either \(r\) or \(h\)