Edexcel Unit 04 Outcome 2 t2

7/28/2019 Edexcel Unit 04 Outcome 2 t2

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EDEXCEL NATIONAL CERTIFICATE

UNIT 4 MATHEMATICS FOR TECHNICIANSOUTCOME 2

TUTORIAL 2 - MENSURATION

Learning outcomes

On completion of this unit a learner should:1 Know how to use algebraic methods

2 Be able to use trigonometric methods and standard formula to determine areas and volumes

3 Be able to use statistical methods to display data

4 Know how to use elementary calculus techniques.

OUTCOME 2 - Be able to use trigonometric methods and standard forlula to determineareas and volumes.

Circular measure: radian; degree measure to radians and vice versa; angular rotations (multiplesof radians); problems involving areas and angles measured in radians; length of arc of a circle

(s = r ); area of a sector (A = r2)

Triangular measurement: functions (sine, cosine and tangent); sine/cosine wave over onecomplete cycle; graph of tanA as A varies from 0 and 360 (tanA = sin A/cosA); values of the

trigonometric ratios for angles between 0 and 360; periodic properties of the trigonometric

functions; the sine and cosine rule; practical problems e.g. calculation of the phasor sum of two

alternating currents, resolution of forces for a vector diagram

Mensuration: standard formulae to solve surface areas and volumes of regular solidse.g. volume of a cylinder = r

2h,

total surface area of a cylinder = 2rh + r2,

volume of sphere (4/3)r3,

surface area of a sphere = 4 r2,

volume of a cone =(1/3) r2h, curved surface area of cone = rx slant height

D.J.Dunn www.freestudy.co.uk 1


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CIRCLES

If we took a cylinder or disc of diameter D and placed it on a flat surface with point 'P' at the bottom

and rolled it forward exactly one revolution until point 'P' was at the bottom again, the distance

moved along the surface by point 'P' is one circumference. The circumference could be measured by

finding the length of a thin piece of string that just goes around the outside once.

Pythagoras discovered a long time ago that no matter how large the circle, the circumference is

always 3.142 diameters. In other words the ratio of the circumference to the diameter is a constant

and 3.142 is the figure found by rounding off to 3 decimal places. The exact ratio is given thesymbol (pi) and this is a number that has no end to the decimal places so normally we round it off

to 3.142 but your calculator should show even more figures if you press the button labelled .

It follows that the circumference of any circle is C = D or in terms of radius C = 2R

ARCS AND SECTORS

A sector is a fraction of a circle and an arc is the corresponding

fraction of the circumference. You should recall that a radian is the

angle made by an arc of length 1 radius so it follows that there are2 radian in a complete circle.

If the angle of the arc is radian and a full circle has an angle of 2 radian then the length of the arc

is2

Dx x

2

encexcircumfer

2

s === or in terms of radius s = R

The length of an arc is the radius x angle (in radian).

The area of the sector is the same fraction of the area of the circle so circletheofAreax

2

A =

The area of a circle is8

D

4

Dx

2

Aso

4

D 222== or in terms of radius

2

RA

2

=



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AREAS OF BASIC SHAPES

Here without proof are the areas of some standard shapes.

SELF ASSESSMENT EXERCISE No.1

1. Calculate the area of an annulus 75 mm outer diameter and 35 mm inner diameter.

(3456 mm2

)2. Calculate the area of triangle shown

(5.25 m2)

3. The area of the triangle shown is 7.5 m2 What is the height h? (3 m)

4. A sector of a circle of diameter 120 mm has an area of 300 mm2. What is the angle of the

sector?

(9.55o)



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VOLUMES AND SURFACE AREAS

PRISMS

A prism is the result of extruding an area A for a length L so that a square becomes a cube, a circle

becomes a cylinder and triangle becomes a wedge.

A prism can have any cross sectional area. The cross sectional dimensions are the same at all points

along the length. The volume is always V = A L

The surface area can usually be deduced by adding all the separate surfaces together. The curved

surface of a cylinder has an area which would be a rectangle when flattened out with length L and

width D so the area is DL

WORKED EXAMPLE No. 1

What is the volume and surface area of a hollow cylinder outer diameter 60 mm, inner diameter

40 mm and length 200 mm?

SOLUTION

The cross sectional area is A = (602 402)/4 = 1570.8 mm2

V = A L = (1570.8)200 = 314159 mm3

The surface area is the surface area of two cylinders and an annulus at each end.

Outer Surface A = DL = x 60 x 200 = 12000Inner Surface A = dL = x40 x 200 = 8000

Two Annular areas A = 2 x(602 402)/4 = 1000

Total = 21000 = 65973 mm2



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SPHERE

Volume V = D3/6

Surface Area A = D2

CONES AND PYRAMIDS

The volume is always 1/3 of the volume of a prism. A pyramid canhave as many sides but the apex must be on the centre line.

The volume of a cone is V = A h/3 = D2h/12

The surface area of the curved surface is A = D L/2 where L is the length of the sloping

side.

In terms of the vertical height h4

DhL

22+=

The curved area becomes 22 D4h4

DA +=

The volume and surface area of the pyramid must be deduced by working out the base area andworking out the area of each flat side.


WORKED EXAMPLE No. 2

A shape as shown is made from a cylinder 50 mm diameterand 100 mm long with a hemisphere on one end.

What is the volume and surface area?

SOLUTION

Volume of cylinder = D2L/4 = x 50

2x 100/4 = 196350 mm

3

Volume of hemisphere = xvolume of sphere = x D3/6

Volume of hemisphere = x x 503/6 = 32725 mm

3

Total Volume = 229075 mm3

Surface area of cylinder curved surface = DL = x 50 x 100 = 5000

Circular area at end = D2/4 = x 502 /4 = 625

Surface area of hemisphere = x D2 = x 502 = 1250Total Surface Area = 6875 = 21598 mm

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SELF ASSESSMENT EXERCISE No. 2

1. Calculate the volume and surface area of a solid cylinder 1.2 m diameter and 1.5 m tall.

(1.696 m3

and 7.9168 m2)

2. Calculate the volume of a cone with base circle 80 mm diameter and a height of 200mm.

If the cone is cut in half to make a small cone 100 mm high and a frustum 100 mm high, what

would be the volume of the two parts?(335103 mm3, 167552 mm3 and 176552 mm3)

3. Calculate the surface area of a cone 120 mm base diameter and 200 mm vertical height.

(50669 mm2)

4. Calculate the volume and surface area of a prism with the cross section of a right angle trianglewith both perpendicular sides 50 mm long and length 100 mm.(125000 mm3 and 19571 mm2 including the ends)

5. The diagram shows a flat thin rectangular plate Width W and Breadth B with a hole diameter D

in the middle.

Create a formula for the area of the metal A in terms of W, B and D.

What is the area if D = 50 mm, B = 100 mm and W= 60 mm?

6. A hollow ball is made from plastic. The ball has an outside diameter D and inside diameter d.

Write down the formula for the volume of plastic used to make it. If the outer diameter is 200 mm and the wall is 2 mm thick, determine the volume of plastic

and the outer surface area.



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7. A rectangular block has a length B, width W and thickness T as shown with a hole diameter Dthrough it.

Write down the formula for the volume 'V' of the block in terms of B, W ,T and D

Calculate the volume when D = 50 mm, B = 100 mm and W= 60 mm and T = 20 mm.

8. A solid object is made in the shape of a cylinder diameter d and length L with a cone on the endbase diameter d and height h.

Write down the formula for the volume 'V' and surface area 'A' in terms of d, L and h.

If the volume is 500 000 mm3 and cylinder is 60 mm diameter by 100 mm long, what is the

height of the cone?


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Edexcel Unit 04 Outcome 2 t2