Higher June 2025 Paper 3 Q16
16 The diagram shows a solid shape made from a cylinder and a cone.
The cone has a base radius of 8 cm.
The cylinder has a radius of 8 cm.
The vertical height of the cone is three times the height of the cylinder.
The volume of the solid shape is \(640\pi\) cm3
Work out the vertical height of the cone.
You must show all your working. (4)
| Answer | Mark | Mark scheme |
|---|---|---|
| 15 | P1 | for process to find volume of cylinder or volume of cone, eg \(\pi \times 8^2 \times \dfrac{h}{3}\) oe or \(\dfrac{1}{3} \times \pi \times 8^2 \times h\) oe or \(\pi \times 8^2 \times H\) oe or \(\dfrac{1}{3} \times \pi \times 8^2 \times 3H\) oe or \(\pi \times 8^2 \times \dfrac{x}{4}\) oe or \(\dfrac{1}{3} \times \pi \times 8^2 \times \dfrac{3x}{4}\) oe |
| P1 | (dep P1) for setting up an equation in terms of one variable, eg \(\text{``}\pi \times 8^2 \times \dfrac{h}{3}\text{''} + \text{``}\dfrac{1}{3} \times \pi \times 8^2 \times h\text{''} = 640\pi\) oe or \(\text{``}\pi \times 8^2 \times H\text{''} + \text{``}\dfrac{1}{3} \times \pi \times 8^2 \times 3H\text{''} = 640\pi\) oe or \(\text{``}\pi \times 8^2 \times \dfrac{x}{4}\text{''} + \text{``}\dfrac{1}{3} \times \pi \times 8^2 \times \dfrac{3x}{4}\text{''} = 640\pi\) oe or \(\text{``}\pi \times 8^2 \times \dfrac{h}{3}\text{''} = 320\pi\) oe or \(\text{``}\dfrac{1}{3} \times \pi \times 8^2 \times h\text{''} = 320\pi\) oe or \(\text{``}\pi \times 8^2 \times H\text{''} = 320\pi\) oe or \(\text{``}\dfrac{1}{3} \times \pi \times 8^2 \times 3H\text{''} = 320\pi\) oe or \(\text{``}\pi \times 8^2 \times \dfrac{x}{4}\text{''} = 320\pi\) oe or \(\text{``}\dfrac{1}{3} \times \pi \times 8^2 \times \dfrac{3x}{4}\text{''} = 320\pi\) oe | |
| P1 | (dep P2) for process to solve for \(h\) or \(H\) or \(x\) eg \((h =)\ \dfrac{640\pi}{\text{``}\frac{1}{3} \times \pi \times 8^2\text{''} + \text{``}\frac{1}{3} \times \pi \times 8^2\text{''}}\) oe eg \((h =)\ \dfrac{3 \times 640\pi}{\text{``}64\pi\text{''} + \text{``}64\pi\text{''}}\) or \((H =)\ \dfrac{640\pi}{\text{``}\pi \times 8^2\text{''} + \text{``}\frac{1}{3} \times \pi \times 8^2 \times 3\text{''}}\ (= 5)\) oe eg \((H =)\ \dfrac{640\pi}{\text{``}64\pi\text{''} + \text{``}64\pi\text{''}}\ (= 5)\) or \((x =)\ \dfrac{640\pi}{\text{``}\pi \times 8^2 \times \frac{1}{4}\text{''} + \text{``}\frac{1}{3} \times \pi \times 8^2 \times \frac{3}{4}\text{''}}\ (= 20)\) oe eg \((x =)\ \dfrac{640\pi}{\text{``}32\pi\text{''}}\ (= 20)\) | |
| A1 | cao |
Additional guidance
\(h\) = height of the cone
\(H\) = height of the cylinder
\(x\) = total height of the shape
Allow any letter for \(h\), \(H\) and \(x\), does not have to be defined for the award of the marks
The award of all marks requires the substitution of \(r = 8\), allow this to be done at a later stage in the question
A correct equation implies the first P1
Allow inconsistent use of \(\pi\) within their equation provided the correct volumes are seen previously
Can be an equation in the form \(ah = p\) or \(bH = q\) or \(cx = r\) where \(a\) or \(b\) or \(c\) is an integer
Award 0 marks for a correct answer with no (or incorrect) supportive working