Higher November 2024 Paper 3 Q19
19 The diagram shows a cone.
The radius of the base of the cone is \(\dfrac{3}{4}\) of the height of the cone.
The total surface area of the cone is \(54\pi\) cm2
Work out the height of the cone. (4)
| Answer | Mark | Mark scheme |
|---|---|---|
| 6 | P1 | for starting process, by defining height, radius and using Pythagoras to form an equation for the slant height \(l\) eg height \(= h\), radius \(= \frac{3}{4}h\) and \(l^2 = h^2 + \left(\dfrac{3h}{4}\right)^2 \left(= \dfrac{25}{16}h^2\right)\) or \((l =) \sqrt{h^2 + \left(\dfrac{3h}{4}\right)^2} \left(= \dfrac{5}{4}h\right)\) oe eg \(r = 3x\) and \(h = 4x\) and \(l^2 = (3x)^2 + (4x)^2\) |
| P1 | (dep P1) for process to form a correct expression for the curved surface area in terms of a single variable, eg \(\pi \times \dfrac{3}{4}h \times \text{``}\dfrac{5}{4}h\text{''} \left(= \dfrac{15}{16}\pi h^2\right)\) or \(\pi \times 3x \times \text{``}5x\text{''}\) where \(h = 4x\) | |
| P1 | for forming and simplifying a correct equation to find height, eg \(\dfrac{24\pi}{16}h^2 = 54\pi\) | |
| A1 | cao |
Additional guidance
Can use any other letter than \(h\) provided it is defined
eg height \(= x\)
May include area of circle
eg
\(\pi \times \left(\dfrac{3}{4}h\right)^2 + \pi \times \dfrac{3}{4}h \times \text{``}\dfrac{5}{4}h\text{''} \left(= \dfrac{24}{16}\pi h^2\right)\)