Higher June 2019 Unit 1 Q16
16. The diagram shows a cylinder.
The cylinder has a base of radius \(r\) and a height of \(\dfrac{r}{6}\).
Diagram not drawn to scale
A sphere has radius \(R\).
The volume of the sphere is equal to the volume of the cylinder.
Find \(R\) in terms of \(r\).
Give your answer in its simplest form. [4]
From the formula list: Volume of sphere \(= \dfrac{4}{3}\pi r^3\)
| Scheme | Marks | Comments |
|---|---|---|
| \(\dfrac{4\pi R^3}{3} = \dfrac{\pi r^3}{6}\) | M2 | Equating volumes Award M1 for sight of: (Volume of cylinder =) \(\pi r^2 \times r/6\) or equivalent \(\dfrac{4\pi r^3}{3} = \dfrac{\pi r^3}{6}\) is awarded M1. |
| \(24R^3 = 3r^3\) or \(R = \sqrt[3]{(\pi r^3/6)/(4\pi/3)}\) or \(R^3 = (\pi r^3/6)/(4\pi/3)\) or equivalent | m1 | Award m1 for clearing fractions AND cancelling \(\pi\) or for isolating \(R\) or for isolating \(R^3\). FT if M1 awarded and if equivalent difficulty |
| \(R = \dfrac{r}{2}\) | A1 | CAO |