CBSE Class 10 - Mathematics - Surface areas and volumes - Exercise 13.3 Solutions

1. A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm. Find the height of the cylinder.

2. Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.

3. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform.

4. A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.

5. A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

6. How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?

Factors and multiples - 8

In the previous post we discussed about the multiples of a number. We hope that you have solved the given practice problems. Did you realize that the tables of various numbers are nothing but the first 10 multiples of the given number?

 Multiples and LCM

Example:

Table of 2 i.e. First 10 multiples of 2	Table of 7 i.e. First 10 multiples of 7	Table of 13 i.e. First 10 multiples of 13
2 x 1 = 2	7 x 1 = 7	13 x 1 = 13
2 x 2 = 4	7 x 2 = 14	13 x 2 = 26
2 x 3 = 6	7 x 3 = 21	13 x 3 = 39
2 x 4 = 8	7 x 4 = 28	13 x 4 = 52
2 x 5 = 10	7 x 5 = 35	13 x 5 = 65
2 x 6 = 12	7 x 6 = 42	13 x 6 = 78
2 x 7 = 14	7 x 7 = 49	13 x 7 = 91
2 x 8 = 16	7 x 8 = 56	13 x 8 = 104
2 x 9 = 18	7 x 9 = 63	13 x 9 = 113
2 x 10 = 20	7 x 10 = 70	13 x 10 = 130

Let’s explore the concept of multiple further.

Write down first 10 multiples of number 2 and 3.

First 10 multiples of 2	First 10 multiples of 3
2 x 1 = 2	3 x 1 = 3
2 x 2 = 4	3 x 2 = 6
2 x 3 = 6	3 x 3 = 9
2 x 4 = 8	3 x 4 = 12
2 x 5 = 10	3 x 5 = 15
2 x 6 = 12	3 x 6 = 18
2 x 7 = 14	3 x 7 = 21
2 x 8 = 16	3 x 8 = 24
2 x 9 = 18	3 x 9 = 27
2 x 10 = 20	3 x 10 = 30

From the table above we observe that within first 10 multiples of each 2 & 3, the common multiples are 6, 12 and 18. Here the smallest common multiple is 6.  

We know that a number has infinitely many multiple. Hence there can be many common multiples between any two numbers.

Let’s consider some more examples.

Multiples of 4 and 6

First 10 multiples of 4	First 10 multiples of 6
4 x 1 = 4	6 x 1 = 6
4 x 2 = 8	6 x 2 = 12
4 x 3 = 12	6 x 3 = 18
4 x 4 = 16	6 x 4 = 24
4 x 5 = 20	6 x 5 = 30
4 x 6 = 24	6 x 6 = 36
4 x 7 = 28	6 x 7 = 42
4 x 8 = 32	6 x 8 = 48
4 x 9 = 36	6 x 9 = 54
4 x 10 = 40	6 x 10 = 60

From the first 10 multiples of 4 and 6, the common multiples are 12, 24 and 36. The smallest common multiple is 12.

Multiples of 6, 9 and 12

First 10 multiples of 6	First 10 multiples of 9	First 10 multiples of 12
6 x 1 = 6	9 x 1 = 9	12 x 1 = 12
6 x 2 = 12	9 x 2 = 18	12 x 2 = 24
6 x 3 = 18	9 x 3 = 27	12 x 3 = 36
6 x 4 = 24	9 x 4 = 36	12 x 4 = 48
6 x 5 = 30	9 x 5 = 45	12 x 5 = 60
6 x 6 = 36	9 x 6 = 54	12 x 6 = 72
6 x 7 = 42	9 x 7 = 63	12 x 7 = 84
6 x 8 = 48	9 x 8 = 72	12 x 8 = 96
6 x 9 = 54	9 x 9 = 81	12 x 9 = 108
6 x 10 = 60	9 x 10 = 90	12 x 10 = 120

From the first 10 multiples of 6, 9 and 12, the common multiple among all is 36 which is the smallest common multiple as well.

The smallest common multiple of given numbers is called Least common multiple (LCM).

From the given examples:

·         LCM of 2 and 3 is 6.  

·         LCM of 4 and 6 is 12.

·         LCM of 6, 9 and 12 is 36.

Practice time:

Write first 10 multiples of each of the following and find the LCM.

         i.        3, 7

        ii.        5, 8

       iii.        12, 15

      iv.        16, 80

       v.        16, 24