4MA0 Higher January 2016 Paper 3 Q19
19 The diagram shows a sphere and a cone.
Diagram NOT accurately drawn
The cone has height \(h\) cm.
The radius of the base of the cone is 3 times the radius of the sphere.
Given that the volume of the sphere is equal to the volume of the cone, find an expression for the radius of the sphere in terms of \(h\).
Give your expression in its simplest form.
(3)
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{3}\pi(3r)^2h\) | M1 |
| \(\dfrac{4}{3}\pi r^3 = \dfrac{1}{3}\pi(3r)^2h\) | M1 |
| \(\dfrac{9h}{4}\) oe | A1 |
| (3) |
Notes
M1: or \(\dfrac{4}{3}\pi\left(\dfrac{y}{3}\right)^3\)
Condone missing brackets around \(3r\) or \(\dfrac{y}{3}\) for this mark only
Allow use of 3.14 consistently (or other correct value)
M1: or \(\dfrac{4}{3}\pi\left(\dfrac{y}{3}\right)^3 = \dfrac{1}{3}\pi y^2h\)
A1: for \(\dfrac{9h}{4}\) oe or \(r = \dfrac{9h}{4}\)
NB : \(h = \dfrac{4r}{9}\) scores M2 A0