Class 09 To draw histograms for classes of equal widths and varying widths.

 Activity 32

   OBJECTIVE                                                                    

To draw histograms for classes of equal widths and varying widths.

MATERIAL REQUIRED

Graph paper, geometry box, sketch pens, scissors, adhesive, cardboard.

 METHOD OF CONSTRUCTION

 1. Collect data from day to day life such as weights of students in a class and make a frequency distribution table.

 Case I : For classes of equal widths 

Class	a-b	b-c	c-d	d-e	e-f
Frequency	f 1	f 2	f3	f4	f5

 Case II : For classes of varying widths

 Here : d – f = 2 (a – b) 

Class	a-b		b-c		c-d		d-f
	(width x)		(width x)		(width x)		(width 2x)
Frequency	f1		f 2		f3		f 4
Modified frequency	f		f		f		F′ =	f4
		1		2		3
							2

 2.   Take a graph paper ( 20 cm × 20 cm) and paste it on a cardboard.

 3.   Draw two perpendicular axes X′OX and YOY′ on the graph paper.

 Mark classes on x-axis and frequencies on y-axis at equal distances as shown in Fig. 1

5.   On intervals (a-b), (b-c), (c-d), (d-e), (e- f), draw rectangles of equal widths and of heights f₁, f₂, f₃, f₄ and f₅, respectively, as shown in Fig. 2.

 6.   On intervals (a-b), (b-c), (c-d), (d-f), draw rectangles of heights f₁, f₂, f₃, and F ' as shown in Fig. 3.

 DEMONSTRATION

 1.   Different numerical values can be taken for a, b, c, d, e and f.

 2.   With these numerical values, histograms of equal widths and varying widths can be drawn.

 OBSERVATION

 Case I

 1. The intervals are 

a-b =

.................,

b-c =

.................,

c-d =

.................,

.................

d-e =	................., e– f	= .................
2. f1 = .................	,  f2 =	.................		,	f3 =	.................	,
f4 = .................	,  f5 =	.................
Case II
1. a-b =	.................,	b-c = .................			,	c-d = .................	,
d-f = .................	,
2. f1 = .................	,	f2 = .................			,	f3 = .................	,
f4 =	,	F′ =	f4	=
			.................
		2

APPLICATION

 Histograms are used in presenting large data in a concise form pictorially.

Class 09 To obtain the formula for the surface area of a sphere

 Activity 31 

OBJECTIVE	MATERIAL REQUIRED
To obtain the formula for the surface area	A ball, cardboard/wooden strips,
of a sphere.	thick sheet of paper, ruler, cutter,
	string, measuring tape, adhesive.
METHOD OF CONSTRUCTION

 1.   Take a spherical ball and find its diameter by placing it between two vertical boards (or wooden strips) [see Fig. 1]. Denote the diameter as d.

 2.   Mark the topmost part of ball and fix a pin [see Fig. 2].

 3.   Taking support of pin, wrap the ball (spirally) with string completely, so that on the ball no space is left uncovered [see Fig. 2].

 4.   Mark the starting and finishing points on the string, measure the length between these two marks and denote it by l. Slowly, unwind the string from the surface of ball.

 5. On the thick sheet of paper, draw 4 circles of radius ‘r’ (radius equal to the radius of ball).

6.   Start filling the circles [see Fig. 3] one by one with string that you have wound around the ball.

 Fig. 3

 DEMONSTRATION

 Let the length of string which covers a circle (radius r) be denoted by a.

 The string which had completely covered the surface area of ball has been used completely to fill the region of four circles (all of the same radius as of ball or sphere).

 This suggests:

 Length of string needed to cover sphere of radius r = 4 × length of string needed to cover one circle

 i.e.,       l = 4a

 or, surface area of sphere = 4 × area of a circle of radius r

 So, surface area of a sphere = 4πr²

OBSERVATION

Diameter d of the spherical ball =................ units

radius r =................ units

Length l of string used to cover ball = ................ units

Length a of string used to cover one circle =............... units

So         l = 4 × ____

Surface area of a sphere of radius r = 4 × Area of a circle of radius _____ = 4πr².

 APPLICATION

This result is useful in finding the cost of painting, repairing, constructing spherical and hemispherical objects.

Note: •    Measure diameter of ball carefully.

 •    Wrap the ball completely so that no space is left uncovered.

 Thinner the string more is the accuracy.

Class 09 To find a formula for the curved surface area of a right circular cylinder,experimentally.

 Activity 30 

OBJECTIVE	MATERIAL REQUIRED
To find a formula for the curved surface	Coloured chart paper, cellotape,
area  of  a  right  circular  cylinder,	ruler.
experimentally.

  METHOD OF CONSTRUCTION

 1.   Take a rectangular chart paper of length l units and breadth b units [see Fig. 1].

 Fold this paper along its breadth and join the two ends by using cellotape and obtain a cylinder as shown in Fig. 2.

DEMONSTRATION

 1.   Length of the rectangular paper = l = circumference of the base of the cylinder = 2πr, where r is the radius of the cylinder.

 2.   Breadth of the rectangular paper = b = height (h) of the cylinder.

 3.   The curved surface area of the cylinder is equal to the area of the rectangle = l × b = 2πr × h = 2πrh square units.

OBSERVATION

 On actual measurement :

 l = ....................,                         b = .....................,

 2π r = l = ....................,         h = b = ....................,

 Area of the rectangular paper = l × b = .................

 Therefore, curved surface area of the cylinder = 2πrh.

 APPLICATION

 This result can be used in finding the material used in making cylindrical containers, i.e., powder tins, drums, oil tanks used in industrial units, overhead water tanks, etc.