# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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Number of People (x) Tally Frequency (f)

0 llll 4

1 III 3

2 Ufl I 6

3 nu un 1 11

4 III 3

5 1 1

Total 28

Reference: Statistics. FREQUENCY CURVE

Reference: Frequency Polygon.

210 FREQUENCY POLYGON

FREQUENCY DISTRIBUTION

References: Frequency, Bar Graph.

FREQUENCY POLYGON

This is a graph that is related to a histogram. The meaning of a frequency polygon will be brought out in the following example.

Nathan collected data about the weight of students in his class of 30 students. He split the weights into class intervals of 5 kg and recorded the data in a frequency table as shown. The class interval 40- means a weight of 40-45 kg, which includes 40 kg, but does not include 45 kg. Nathan used the frequency table to draw a histogram of the results. He joined the midpoints of the tops of the columns with a series of straight-line segments to obtain the frequency polygon shown in the figure.

Weight in kg (x) Frequency (f)

40- 1

45- 3

50- 4

55- 9

60- 6

65- 5

70-75 2

Total 30

Weight of students in kg

The columns of the histogram are not part of the frequency polygon and they can be removed so that the graph of the straight-line segments is left. The frequency polygon has its ends extended to reach the horizontal axis. To draw a frequency polygon the data should be continuous.

If the midpoints of the tops of the columns are joined with a smooth curve instead of a series of straight-line segments, the graph is called a frequency curve.

References: Class Interval, Continuous Data, Histogram.

FRUSTUM 211

FREQUENCY TABLE

References: Arithmetic Mean, Bar Graph, Frequency.

FRUSTUM

A frustum is a solid shape that is left when the top of a cone or a pyramid is sliced off by a plane parallel to the base of the cone. In figure a, the cone is an upright one, called a right cone, and the plane of the slice is a circle. It is sometimes called a truncated cone.

The net of a hollow frustum of a cone, without the two circles for its ends, is shown in figure a. The net is a part of the sector of a circle, and the completion of the sector is shown by dashed lines.

The pattern of a girlâ€™s skirt is a net of the frustum of a cone. The surface area of the frustum of the cone (not including the two circular ends) is (see figure b)

(b)

Area = TtS(R + r)

The volume of the solid frustum of the cone is

Example. Helen is cutting out a pattern for a skirt she is making. Her waist measurement is 80 cm, the hem of her skirt is to be 140 cm, and the length of the skirt is 60 cm. What is the area of the pattern for her skirt?

212 FRUSTUM

Solution. The left-hand side of figure c shows the net of the skirt, and the right-hand side is a three-dimensional sketch of the finished skirt. The formula for the area of the skirt is Area = nS(R + r). To evaluate this area, we require the values of r and R, neither of which is known. The value of 5 is 60 cm. The ensuing working shows how to find the values of r and R.

The formula for the circumference of a circle is C = 2ttR. The waist circle in the three-dimensional sketch has a circumference of 80 cm. Write

80 = 2irr

_ 80 2jt

r = 12.7 cm (to 1 dp) Similarly

R

140

2tt

R =22.3 cm (to 1 dp)

Now write

A = TtS(R + r)

A = ?r x 60 x (22.282 + 12.732)

Substituting C = 80 and r is the radius. Rearranging the equation to make r the subject Using the value of it in the calculator

Which is the radius of the hem of the skirt Using the value of n in the calculator.

Formula for the area of the skirt

Substituting 5 = 60, R = 22.282, and r = 12.732

A = 6600 to nearest whole number The area of the pattern is 6600 cm2, or 0.66 m2 (1 m2 = 10,000 cm2).

Frustum of a Square-based Pyramid The volume of the frustum is given by (see figure d)

Volume = \h(a2 + ab + b2) where a, b, and h are shown in figure d.

FURLONG 213

The right-hand side of the figure is the net of the frustum of the square-based pyramid, which is made up of two squares and four trapeziums.

References: Circle, Cone, Net, Pyramid, Sector of a Circle, Square, Trapezium. FUNCTION

A full description of function is given in the entry Correspondence. The relation tree in the figure shows all the subsets of relations, and two types of relations are functions

Reference: Correspondence. FURLONG

Reference: Imperial System of Units.

G

GALLON

Reference: Imperial System of Units.

GEARS

A gear is a wheel with teeth. Two gear wheels are used together to change a turning speed and the direction of the turning. Two gear wheels are used on a bicycle, one at the pedal and the other at the back wheel. A chain connects the gear wheels, which ensures the wheels turn in the same direction.

Example. A chain connects the gear wheels shown in figure a. For one revolution of the larger gear wheel how many times does the smaller gear wheel turn?

(a)

Solution. The larger gear wheel has 12 teeth and the smaller gear wheel has 8 teeth. Write

The ratio of teeth = 12:8 Larger wheel to smaller wheel

= 3:2 Canceling down the ratio

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