Numeracy Higher November 2019 Unit 1 Q10
10. Nicola makes party hats for young children.
The hats are in the shape of cones.
Nicola’s design is shown below.
Diagram not drawn to scale
Nicola makes the hats by folding pieces of card that are in the shape of sectors of a circle.
The hats are formed when the straight edges meet.
Diagram not drawn to scale
(a) Calculate the sector angle Nicola uses in her design. [4]
(b)
Nicola also makes party hats for adults.
They are made from pieces of card that are sectors of a circle, with radius 24 cm and sector angle 150º.
Nicola cuts each sector of a circle from a sheet of rectangular card that measures 24 cm by 50 cm.
Calculate the area of card that is wasted from each rectangular sheet.
Give your answer in terms of π in its simplest form. [4]
| Scheme | Marks | Comments |
|---|---|---|
| Sight of \(\dfrac{\text{Angle}}{360}\times 2\pi\times 20\) | B1 | Accept numerical values of π used |
| \(\dfrac{\text{Angle}}{360}\times 2\pi\times 20 = \pi\times 15\) or equivalent | M1 | |
| (Angle =) \(\dfrac{\pi\times 15\times 360}{2\pi\times 20}\) or equivalent | m1 | This m1 implies the previous B1M1 |
= 135 (º) | A1 | CAO |
Alternative
Alternative method 1:
| Scheme | Marks | Comments |
|---|---|---|
| Sight of \(\dfrac{\text{Angle}}{360}\times \pi\times 20^2\) | B1 | Accept numerical values of π used |
| \(\dfrac{\text{Angle}}{360}\times \pi\times 20^2 = \pi\times 7.5\times 20\) or equivalent | M1 | |
| (Angle =) \(\dfrac{\pi\times 7.5\times 20\times 360}{\pi\times 20^2}\) | m1 | |
= 135 (º) | A1 | CAO |
Alternative
Alternative method 2:
| Scheme | Marks | Comments |
|---|---|---|
| Sight of circumferences 15π AND 40π OR Sight of appropriate areas 150π AND 400π | B1 | |
| Sight of \(\dfrac{3}{8}\) or \(\dfrac{15(\pi)}{40(\pi)}\) or \(\dfrac{150(\pi)}{400(\pi)}\) OR 3:8 or 15(0π) : 40(0π) | B1 | This B1 implies the first B1 |
| \(\dfrac{3}{8}\times 360\) or equivalent | M1 | This implies the previous B1B1 |
= 135 (º) | A1 | CAO |
| Scheme | Marks | Comments |
|---|---|---|
| (Area of sector =) \(\dfrac{150}{360}\times\pi\times 24^2\) | M1 | Accept numerical values of π used for a possible M1m1 only |
| \(\dfrac{5}{12}\times\pi\times 24^2\) OR 5 × π × 2 × 24 OR \(\dfrac{150}{5}\times\pi\times 8\) OR \(\dfrac{8640(0)\,\pi}{36(0)}\) | m1 | |
= 240π | A1 | May be embedded or implied in final answer |
| (Area wasted in cm2 =) 1200 − 240π or 240(5 − π) | B1 | Mark final answer, but do not penalise incorrect attempts to factorise a correct answer Accept partially factorised final answer FT ‘their 240π’ provided M1 awarded AND ‘their 240π’ is in its simplest form AND is a multiple of π |