Chapter 3: Squares and Square Roots & Cubes and Cube Roots

Solution:-

From the question it is given that,

Length of rectangle = 6.4 m

Breadth of rectangle = 2.5 m

Area of rectangle = length × breadth

= 6.4 × 2.5

= 16 m²

And also it’s given in the question i.e. area of square is equal to the area of rectangle.

Let us assume area of square be ‘a’.

Area of square = Area of rectangle

a × a = 16

a² = 16

By taking square root on both side

a = √16

a = √(4 × 4)

a = √4²

a = 4m²

We know that, area of square = side²

4 = side²

By taking square root on both side

side = √4

side = √(2 × 2)

side = √2²

side = 2m

Solution:-

From the question it is given that,

Difference of two perfect cubes = 189

The cube root of the smaller of the two numbers = 3

So, cube of smaller number = 3³

= 3 × 3 × 3

= 27

Let us assume the cube root of larger number be a³

Then,

As per the condition given in the question,

a³ – 27 = 189

a³ = 189 + 27

a³ =216

By taking cube root on both side,

a = ³√216

a = ³√(6 × 6 × 6)

a = ³√(6³)

a = 6

∴The cube root of the larger number is 6.

Solution:-

From the question it is given that total number of plants arranged = 1024

Let us assume number of plants in each row be ‘a’

As per the condition given in the question,

Number of plants in a row is the same as the number of rows i.e. ‘a’

Then,

Total number of plants = a × a

1024 = a²

By taking square root on both side,

a = √(1024)

a = √(32 × 32)

a = √(32)²

a = 32

∴The number of plants in each row is 32.

Solution:-

From the question it is given that total number of seats = 2704

Let us assume number of seats in each row be ‘a’

As per the condition given in the question,

Number of rows is equal to the number of seats in each row i.e. ‘a’

Then,

Total number of seats = a × a

2704 = a²

By taking square root on both side,

a = √(2704)

a = √(52 × 52)

a = √(52)²

a = 52

∴The number of seats in each row is 52.

Solution:-

From the question it is given that total number of soldiers = 7500

By using long division method we can able to find number of soldiers left out,

∴104 soldiers were left out.

Solution:-

From the question it is given that,

Total number of students were sitting in a lecture room = 8649

Let us assume number of students in each row be ‘a’

Then,

Total number of students = a × a

8649 = a²

By taking square root on both side,

a = √(8649)

a = √(93 × 93)

a = √(93)²

a = 93

∴The number of students in each row is 93.

Solution:-

From above the figure,

Consider the triangle PQR.

We know that, Pythagoras theorem,

PR² = PQ² + QR²

PR² = 12² + 35²

PR² = 144 + 1225

PR² = 1369

PR = √1369

PR = 37m

∴Rahul walked 37m distance for returning to his house.

Solution:-

Let us consider the right angle triangle PQR

Given,

Long ladder i.e. RQ = 5.5 m

The ladder reaches the wall to a height i.e. PR = 4.4 m

From Pythagoras theorem,

RQ² = RP² + PQ²

5.5² = 4.4² + PQ²

PQ² = 5.5² – 4.4²

PQ = √(5.5² – 4.4²)

PQ = √(30.25 – 19.36)

PQ = √10.89

PQ = 3.3

∴The distance between the wall and the foot of the ladder is 3.3 m.

Solution:-

From the question it is given that,

Total coins the advisor get at the end = 1 + 3 + 5 +…..

The coins order is in the series of odd natural number,

So, the number of terms (n) = 30

Then,

Sum of odd natural numbers = n²

= 30²

= 900

∴ Total coins the advisor get at the end is 900.

Solution:-

Let us assume the three number be 2a, 3a, 5a

Then,

Given, sum of squares of three numbers is 608

i.e. (2a)² + (3a)² + (5a)² = 608

4a² + 9a² + 25a² = 608

38a² = 608

a² = 608/38

a² = 16

a = √16

a = √(4 × 4)

a = √(4²)

a = 4

∴The numbers are, 2a = 2 × 4 = 8

3a = 3 × 4 = 12

5a = 5× 4 = 20