Higher November 2022 Paper 3 Q23
23 Alec makes a bowl for dog food from a solid wooden cone.
The sketches show how the bowl is made.
The cone has radius 9 cm and perpendicular height 30 cm
A smaller cone, with radius 6 cm, is removed.
| Volume of a cone \(= \dfrac{1}{3}\pi r^2 h\) where \(r\) is the radius and \(h\) is the perpendicular height |
A hemisphere with radius 6 cm is then removed.
| Volume of a hemisphere \(= \dfrac{2}{3}\pi r^3 \quad\) where \(r\) is the radius |
Work out the volume of the remaining wood that forms the bowl. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{1}{3} \times 9^2 \times 30 \times \pi\) or \(810\pi\) or [2543, 2545.1] | M1 | oe |
| \(\dfrac{2}{3} \times 6^3 \times \pi\) or \(144\pi\) or [452.1, 452.5] | M1 | |
| \(30 \times \dfrac{6}{9}\) or 20 or \(\left(\dfrac{6}{9}\right)^3\) | M1 | oe implied by \(240\pi\) or [753.6, 754.1] |
| \(\dfrac{1}{3} \times 9^2 \times 30 \times \pi - \dfrac{1}{3} \times 6^2 \times\) their \(20 \times \pi\) or \(\dfrac{1}{3} \times 9^2 \times 30 \times \pi - \dfrac{1}{3} \times 9^2 \times 30 \times \left(\dfrac{6}{9}\right)^3 \times \pi\) or \(810\pi - 240\pi\) or their [2543, 2545.1] \(-\) their [753.6, 754.1] or \(570\pi\) or [1788.9, 1791.5] | M1dep | dep on 1st and 3rd M1 |
| \(426\pi\) or [1336, 1339.4] | A1 |
Additional guidance
All values may be seen on diagrams