Higher June 2024 Paper 3 Q21
21 A cylinder, C, and a sphere, S, each have radius \(r\)
C has height \(h\)
| Volume of a sphere \(= \dfrac{4}{3}\pi r^3\) where \(r\) is the radius |
(a) volume of C \(=\) volume of S
Work out the ratio \(\quad r : h\)
You must show your working. [3 marks]
(b) A different cylinder has radius \(3r\) and height \(2h\).
How many times bigger is the volume of this cylinder than the volume of C? [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{4}{3}\pi r^3 = \pi r^2 h\) | M1 | |
| \(\dfrac{4}{3}r = h\) or \(4r = 3h\) | M1dep | oe equation with \(\pi\) and \(r^2\) cancelled |
| 3 : 4 with M2 awarded | A1 | oe ratio eg \(\dfrac{3}{4} : 1\) or \(1 : \dfrac{4}{3}\) accept 1.33 or better for \(\dfrac{4}{3}\) |
| Alternative method 2 | ||
| \(\dfrac{4}{3}\pi r^3 = \pi r^2 h\) or substitution of the same value of \(r\) into \(\dfrac{4}{3}\pi r^3\) and \(\pi r^2 h\) | M1 | the substitution must be shown |
| Substitution of the same value of \(r\) into \(\dfrac{4}{3}\pi r^3\) and \(\pi r^2 h\) and correct value of \(h\) for their value of \(r\) | A1 | the substitution must be shown their \(h\) should be exactly \(\dfrac{4}{3} \times\) their \(r\) eg \(r = 2\) and \(h = \dfrac{8}{3}\) (oe fraction) do not allow rounded values |
| 3 : 4 with M1A1 awarded | A1 | oe ratio eg \(\dfrac{3}{4} : 1\) or \(1 : \dfrac{4}{3}\) accept 1.33 or better for \(\dfrac{4}{3}\) |
Additional guidance
Accept \(h : r = 4 : 3\) for final mark with M2 or M1A1 awarded
| Answer | Mark | Comments |
|---|---|---|
| \((\pi)(3r)^2(2h)\) or \(3^2 \times 2\) | M1 | oe ft their formula for a cylinder from part (a) in the form \(k\pi r^2 h\) with \(k\) as a positive constant |
| 18 | A1 |
Additional guidance
| Answer 18 from choosing values for \(r\) and \(h\) eg \(\pi \times 3^2 \times 4 = 36\pi\) and \(\pi \times 9^2 \times 8 = 648\pi\) and \(648\pi \div 36\pi = 18\) | M1A1 |
| Answer 18 from rounding a decimal | M0A0 |