Chapter 10 Knowledge & Understanding 1) A circle, centre O ... · Chapter 10 Knowledge & Understanding Page 1 1) A circle, centre O, has radius 36 centimetres. Part of this circle

Chapter 10 Knowledge & Understanding

Page 1

1) A circle, centre O, has radius 36 centimetres.

Part of this circle is shown.

Angle AOB = 140 º.

Calculate the length of arc AB. (3)

2) A fan has four identical plastic blades.

Each blade is a sector of a circle of radius 5 centimetres.

The angle at the centre of each sector is 64o .

Calculate the total area of plastic required to make the blades. (3)

3) Figure 1 shows the circular cross-section of a tunnel with a horizontal floor.

In figure 2, AB represents the floor. AB is 2.4 metres.

The radius, OA, of the cross-section is 2.5 metres.

Find the height of the tunnel. (4)


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4)

The diagram shows a sector of a circle, centre C .

Angle ABC is 160o and the radius of the circle is 30 cm.

Calculate the length of the arc AB . (3)

5) Figure 1 shows a road bridge.

The curved part of the bridge is formed from an arc of a circle, centre O, as shown

in figure 2.

OA and OB are radii of length 170 metres.

The height of the middle of the bridge above its ends is 28 metres as shown in

figure 2.

Calculate the horizontal distance, AB .

Do not use a scale drawing. (4)


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6) A sensor in a security system covers a horizontal area in the shape of a sector of a

circle of radius 15 m.

The area of the sector is 200 square metres.

Find the length of the arc of the sector. (4)

7) The diagram shows a tent.

The shape of the material used to make the tent is a sector of a circle as

shown below.

O is the centre of the circle.

OA and OB are radii of length

3 metres.

Angle AOB is 240o.

Calculate the area of this piece of material. (3)


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8)

A fan is made from a sector of a circle, centre C, where the angle o120ACB

and the radius of the circle is 20 cm. Calculate the length of the arc AB. (3)

9) Figure 1 shows part of an athletics field which is used for the discus event.

The discus is thrown from within a circle of radius 1.25 metres and must land

in the part of the field shaded in the diagram below. (Figure 2)

OAB and OCD are sectors of circle, centre O with radii 1.25OB metres

and 100OC metres. Angle o40AOB .

Calculate the area of the part of the field where the discus must land. (4)

Chapter 10 Reasoning & Enquiry

Page 5

1) A spiral staircase is being designed.

Each step is made from a sector of a circle as shown.

The radius is 1·2 metres.

Angle BAC is 42°.

For the staircase to pass safety regulations, the arc BC must be at least

0·9 metres.

Will the staircase pass safety regulations? (4)

2) A pipe has water in it as shown.

The depth of the water is 5 centimetres.

The width of the water surface, AB, is 18 centimetres.

Calculate r, the radius of the pipe. (3)


Page 6

3) A cone is formed from a paper circle with a sector removed as shown.

The radius of the paper circle is 30 cm.

Angle AOB is 100 º.

(a) Calculate the area of paper used to make the cone. K.U. (3)

(b) Calculate the circumference of the base of the cone. (3)

4) A circle, centre the origin is shown.

P is the point (8, 1).

a) Calculate the length of OP.

The diagram also shows a tangent from P which touches the circle at T.

The radius of the circle is 5 units.

b) Calculate the length of PT. (4)

5) Contestants in a quiz have 25 seconds to answer a question.

This time is indicated on the clock.

The tip of the clock hand moves through the arc

AB as shown.

a) Calculate the size of angle AOB K.U.(1)

b) The length of arc AB is 120 centimetres.

Calculate the length of the clock hand. (4)


Page 7

6) The diagram shows water lying in a length of roof guttering.

The cross-section of the guttering is a semi-circle with diameter 10 centimetres.

The water surface is 8 centimetres wide.

Calculate the depth, d , of water in the guttering. (4)

7) A set of scales has a circular dial.

The pointer is 9 centimetres long.

The tip of the pointer moves through an arc of 2 centimetres for each 100 grams of

weight on the scales.

A parcel, placed on the scales, moves the pointer through an angle of 284o.

Calculate the weight of the parcel. (4)

8) A badge is made from a circle of radius 5 centimetres.

Segments are taken off the top and the bottom of the circle as shown.

The straight edges are parallel.


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The badge measures 7 centimetres from the top to the bottom.

The top is 8 centimetres wide.

Calculate the width of the base. (5)

9) An oil tank has a circular cross-section of radius 2.1 metres.

It is filled to a depth of 3.4 metres.

a) Calculate x, the width in metres of the oil surface. K.U.(3)

b) What other depth of oil would give the same surface width ? (1)

10) The curved part of a doorway is an arc of a circle with radius

500 millimetres and centre C.

The height of the doorway to the top of the arc is 2000 millimetres.

The vertical edge of the doorway is 1800 millimetres.

Calculate the width of the doorway. (5)


Page 9

11) A sheep shelter is part of a cylinder

as shown in Figure 1.

It is 6 metres wide and 2 metres high.

The cross-section of the shelter is

a segment of a circle with centre O,


OB is the radius of the circle.

Calculate the length of OB . (4)

12) The diagram shows a table whose top is in the shape of part of a circle

with centre, O, and radius 60 centimetres.

BD is a straight line.

Angle BOD is 90o.

Calculate the perimeter of the table top. (3)

13) The diagram shows the design of an earring.

The earring consists of a circle placed inside

an equilateral triangle.

The sides of the triangle are tangents to the circle.

The radius of the circle is 8 mm.

The distance from the centre of the circle to each vertex of the triangle

is 17 mm. Calculate the perimeter of the triangle. (4)


Page 10

14) The boat on a carnival ride travels along an arc of a circle, centre C .

The boat is attached to C by a rod 6 metres long.

The rod swings from position CA to position CB .

The length of the arc AB is 7 metres.

Find the angle through which the rod swings from position A to position B. (4)

15)

A cylindrical tank, 5 metres long, is used to store a hazardous liquid as

shown in Figure 1.

A dipstick is used to measure the depth of the liquid.

The dipstick passes through the centre, O, of a cross-section of the tank


The diameter of the cross-section of the tank is 2 metres.

To satisfy safe storage conditions, the horizontal surface of the liquid

must not be less than 2 square metres.


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a) Explain why the width of the horizontal surface of the liquid must not

be less than 0.4 metres.

b) The depth of the liquid in the tank is found to be 1.8 metres.

Can this volume be stored safely in the tank ?

Justify your answer. (7)


Page 12

1 ) ST, a vertical pole 2 metres high, is situated at the corner of a rectangular garden,

PQRS . RS is 8 metres long and QR is 12 metres long.

The pole casts a shadow over the garden.

The shadow reaches M , the midpoint of QR .

Calculate the size of the shaded angle TMS . (4)

2) In the diagram,

Angle o34STV

Angle o25VSW

Angle SVT Angle o90SWV

13.1ST centimetres.

Calculate the length of SW . (4)

3) A statue stands at the corner of a square courtyard.

The statue is 4.6 metres high.

The angle of elevation from the opposite corner of the courtyard to the

top of the statue is 8o.

a) Find the distance from the base of the statue to the opposite corner

of the courtyard.

b) Show that the length of the side of the courtyard is approximately

23 metres. (5)


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4)

The diagram above shows a garden hut.

The hut is 3 metres high at one side and 2.5 metres at the other.

Its roof slopes at an angle of 15o to the horizontal.

Calculate the length, S metres of the sloping edge of the roof, giving your

answer to 2 significant figures.


5)

The diagram above shows the design of a swimming pool 50 metres in

length.

The pool is 1 metre deep at one end and its base slopes downwards at an

angle of 3o to the horizontal.

Calculate the depth, d metres , of the other end of the pool, giving your

answer to 2 significant figures.



Page 14

1) The chain of a demolition ball is 12.5 metres long.

When vertical, the end of the chain is 1.5 metres from the ground.

It swings to a maximum height of 2.5 metres above the ground on both sides.

a) At this maximum height, show that the angle ox , which the chain

makes with the vertical, is approximately o23 .

b) Calculate the maximum length of the arc through which the end of the

chain swings. Give your answer to 3 significant figures. (8)

2) The diagram below shows a spotlight at point S , mounted 10 metres directly

above a point P at the front of a stage.

The spotlight swings 45o from the vertical to illuminate another point Q,

also at the front edge of the stage.

Through how many more degrees must the spotlight swing to illuminate a point B,

where Q is the mid-point of PB ? (5)


Page 15

3) a) A lampshade is made in the shape of a cone,

as shown.

The shape of the material used for the

lampshade is a sector of a circle.

The circle has radius 25 centimetres and

the angle of the sector is 280o.

a) Find the area of the sector of the circle. K.U. (3)

Each sector is cut from a rectangular piece of material,

50 centimetres wide.

b) Find, to the nearest centimetre, the minimum

length, L , required for the piece of material.

(4)

4) The Scott family want to build a conservatory as shown below.

The conservatory is to be 3 metres wide.

The height of the conservatory at the lower end

is to be 2 metres and at the higher end 3.5 metres.

To obtain planning permission, the roof must slope

at an angle of 25 2 degrees to the horizontal.

Should planning permission be granted ?

Justify your answer. (4)


Page 16

5) The diagram below shows a ceiling in the shape of a rectangle and a segment

of a circle.

The rectangle measures 8.3 metres by 4.5 metres.

OB and OC are radii of the circle and angle BOC is 130o.

a) Find the length of OB .

A border has to be fitted round the perimeter of the ceiling.

b) Find the length of border required. (7)

Answers To Chapter 10-11

Page 17


1) 88 cm 2) 56 cm2 3) 4.69 m 4) 83.8 cm2 5) 187 m 6) 26.7 cm

7) 18.9 m2 8) 41.9 cm 9) 3490.45 m2


1) No, since 0.88 < 0.9 2) 10.6 cm 3) a) 2041 cm2 b) 136.1 cm 4) a) 8.06

b) 6.3 5) a) 150º b) 45.84 6) 2 cm 7) 2230 g 8) 6 cm 9) a) 3.3 m b) 0.8 m

10) 800 mm 11) 13

4x 12) 367.5 cm 13) 90 mm 14) 66.9º 15) a) Proof

b) Yes it is safe since 3m2 > 2m2


1) 11.3º 2) 6.6 cm 3) a) 32.7 m b) 23.1 m 4) 1.9 5) 3.62 m


1) a) Proof b) 10m 2) 18.4º 3) a) 1526.4 cm2 b) 41.1 cm

4) Yes planning permission should be given since 26.6º is between 23º and 27º

5) a) 2.48 m b) 26.73 m

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Chapter 10 Knowledge & Understanding 1) A circle, centre O ... · Chapter 10 Knowledge & Understanding Page 1 1) A circle, centre O, has radius 36 centimetres. Part of this circle