11. A company produces metal badges to be worn by its employees. The badge is made up of two parts. One part is in the shape of a sector of a circle as shown in the diagram.
Diagram not drawn to scale
(a) The perimeter of the sector is decorated with a coloured edging strip. Calculate the length of edging strip needed to decorate the sector. [3]
(b) The other part is in the shape of a quarter-circle of radius 3 cm.
Diagram not drawn to scale
To make the badge, the two pieces are joined together with the sector in front of the quarter-circle, as shown in the diagram. The badge has a vertical line of symmetry.
Diagram not drawn to scale
The visible surface of the front of the badge is painted. Calculate the area that is painted. [5]
Mark scheme (a)
Scheme
Marks
Comments
(Length of arc) \(\frac{50}{360} \times 2 \times \pi \times 5\)
(Surface area of badge =) \(7 \cdot 06 \ldots + 10 \cdot 91 \ldots - \frac{50}{360} \times \pi \times 3^2\) (3∙925 to 3∙9275)
M1
FT ‘their 7∙06…’ and ‘their 10∙91…’ provided previous M1 awarded
= 14∙0(427…) to 14∙1 OR 161π/36 (cm2)
A1
Needs to come from values that are correct to at least 1 decimal place
Alternative method: B1 for (\(\frac{20}{360} \times \pi \times 3^2\)) 1∙57 to 1∙571 or π/2 (cm2) (may be implied in further working) M1 for \(\frac{50}{360} \times \pi \times 5^2\) A1 for 10∙9(027…) to 10∙91 OR 125π/36 (cm2) M1 for \(10 \cdot 9 \ldots + (2 \times \frac{20}{360} \times \pi \times 3^2)\) FT ‘their 10∙9’ and ‘their 1∙57’ provided previous M1 awarded A1 for 14∙0(427…) to 14∙1 OR 161π/36 (cm2)
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