12. A new athletics stadium is to be built in Alltycapel.
(a) A throwing circle is to be built for the shot put and discus events. There are lines drawn from the centre of the circle. They show the athletes where the boundaries are for their throws. The lines form a sector of the circle. This sector is to be painted, as shown in the diagram.
Diagram not drawn to scale
The radius of the throwing circle is 120 cm.
The area of the sector is \(0{\cdot}08\dot{3}\) of the area of the circle.
(i) Write \(0{\cdot}08\dot{3}\) as a fraction in its simplest form. [3]
(ii) Use your answer to (i) to calculate the area to be painted. Give your answer in terms of π in its simplest form. [2]
(b) A new running track is to be built at the stadium.
Athletes in a 200-metre race run in lanes. The inside line of one of the lanes is shown below.
The inside line consists of:
a straight section of length 90 m,
an arc of a circle with radius 36 m.
The length of this inside line is 200 m.
Diagram not drawn to scale
Show that the value of \(x\) is \(\dfrac{550}{\pi}\). [5]
Mark scheme (a)
(i)
Scheme
Marks
Comments
e.g. 100x = 8.333…, 1000x = 83.333… AND an attempt to subtract both sides
M1
= \(\dfrac{75}{900}\) or \(\dfrac{825}{9900}\) or \(\dfrac{8325}{99900}\) or equivalent
A1
Allow A1 for e.g. 7.5/90
= \(\dfrac{1}{12}\)
A1
FT from M1A0 provided of equivalent difficulty
(ii)
Scheme
Marks
Comments
(Area =) \(\dfrac{1}{12}\) × π × 1202 or equivalent
M1
FT ‘their 1/12’ from (i) throughout
= 1200π (cm2)
A1
If no marks awarded, SC1 for (11/12 × π × 1202 or equivalent =) 13200π (cm2)
Mark scheme (b)
Scheme
Marks
Comments
Sight of \(\dfrac{x}{360}\) × 2 × π × 36 or equivalent
B1
e.g. \(\dfrac{x}{5}\) × π. Accept any symbol for x
\(\dfrac{x}{360}\) × 2×π×36 + 90 = 200 or equivalent
M1
\(\dfrac{x}{360}\) × 2×π×36 = 200 ─ 90 or equivalent
m1
These two m1 marks can be done in any order For isolating the x term
For fully simplifying the fraction correctly OR for isolating x
\(x\) = \(\dfrac{550}{\pi}\)
A1
Needs to come from convincing work from M1m1m1 e.g. \(\dfrac{39600}{72\pi}\) or \(\dfrac{110 \times 360}{72\pi}\)
If no marks awarded or B1 only awarded, then SC1 for \(\dfrac{550 \times 2 \times \pi \times 36}{360\pi}\) or equivalent AND possibly another SC1 for convincing work showing that this simplifies to 110, and that 110 + 90 = 200
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