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Given:

The ratio of radius and height = 4 : 3

Formula used:

The lateral surface area of cone = πrl

The total surface area of cone = πr(l + r)

l² = r² + h²

Where,

r = radius of cone

l = slant height of the cone

Calculation:

Let r = 4 and h = 3

l² = r² + h²

⇒ l² = 4² + 3²

⇒ l = 5

The ratio between lateral surface area and total surface area of the cone

⇒ (πrl)/[ πr(l + r)]

⇒ l/(l + r)

⇒ 5/(5 + 4)

⇒ 5/9

Given:

The perimeter of the base of a right circular cone is 44 cm.

Height of the cone = 24 cm

Formula used:

The circumference of the base of the cone = 2πr

The curved surface area of the cone = πrl

The slant height of the cone (l) = √(h² + r²)

Where r → radius of the cone 

h → height of the cone 

l → slant height of the cone

Calculations:

According to the question,

The circumference of the base of the cone = 44 cm.

⇒ 2πr = 44

⇒ r = 7 cm

The slant height of the cone = √(h2 + r2)

⇒ l = √(242 + 72) = √(625) = 25

So, the curved surface area of the cone = πrl

⇒ (22/7) × 7 × 25 = 550 cm²

∴  The curved surface area of the cone is 550 cm2

Given:

Volume of cone = 154 cm³

Height of cone = 12 cm

Concept:

Using the volume formula, calculate the radius of the cone and then calculate the lateral height.

Formula used:

Volume of cone = (1/3)πr²h

Lateral surface area of cone = πrl

Lateral height (l) = √(r² + h²)

Calculation:

∵ Volume of cone = (1/3)πr²h

⇒ 154 = (1 × 22 × r² × 12)/(3 × 7)

⇒ 154 × 3 × 7 = 22 × r² × 12

⇒ r² = (154 × 3 × 7)/( 22 × 12)

⇒ r² = 49/4

⇒ r = √(49/4) = 7/2

∵ Lateral height (l) = √(r² + h²)

⇒ l = √[(7/2)² + (12)²]

⇒ l = √[144 + (49/4)]

⇒ l = √[(576 + 49)/4]

⇒ l = √(625/4)

⇒ l = 25/2 cm

∴ Lateral surface area of cone = πrl

= (22/7) × (7/2) × (25/2)

= 275/2 cm²

= 137.5 cm²

Given:

Volume of a sphere is 38,808 cm3

Radius of a sphere is twice of the hemisphere.

Formula Used:

Volume of a sphere = 4/3πr³ 

Lateral surface area of a hemisphere = 2πr²

Calculation:

Volume of a sphere = 4/3πr3 

⇒ 4/3π r3 = 38,808

⇒ r3 = (38,808 × 3)/ 4π

⇒ r3 = 9261

⇒ r = 21

As radius of a sphere is twice of a hemisphere.

⇒ radius_(circle) = 2 radius_(hemisphere)

⇒ 21 = 2 radius(hemisphere)

⇒ radius_(hemisphere) = 21/2

Now, Lateral surface area of a hemisphere = 2πr2

⇒ Lateral surface area of a hemisphere = 2π × (21/2)²

⇒ Lateral surface area of a hemisphere = 693

∴ The lateral surface area of a hemisphere is 693 cm².

Given:

Length of room = 26 m

Breadth of room = 15 m

Side of carpet = 14 m 

Formula Used:

Area of a rectangle = Length × Breadth

Area of a square = (Side)² 

Calculation:

Area of the room = 26 × 15 m² 

⇒ 390 m2 

Area of carpet = 14 m2

⇒ 196 m2 

Area in room where carpet is not present = Area of room – Area of carpet laid

⇒ 390 m2  – 196 m2 

⇒ 194 m2 

∴ The area of room which is not carpeted is 194 m².

Given

Base = height/2

Concept

Area of triangle = (1/2) × base × height

Calculation

Let height of triangle is h m

So, base is (h/2) m

Area of triangle = (1/2) × (h/2) × h

⇒ h²/4 m²

Rs. 150 = 100 m²

⇒ Rs. 1 = 100/150

⇒ Rs. 600 = (100/150) × 600 m²

⇒ Rs. 600 = 400 m²

Now,

400 = h² /4

⇒ 1600 = h²

⇒ √1600 = h

⇒ 40 m = h

Base = 40/2

⇒ 20 m

Given:

The volume of each cube is 216 cm cube

Formula Used:

Side = ∛ Volume of cube

Lateral surface area of cuboids = 2 × (l + b) × h

Total surface area of cuboids = 2 × (lb + bh + hl)

Calculation:

When three cubes are place side by side so, the breadth and height of resulting cuboids will be same as cube but length will change

∴ Length of cuboid = 6 + 6 + 6 = 18 cm

So, length = 18 cm, breadth = 6 cm and height = 6 cm

∴ Lateral surface area of cuboids = 2 × (18 + 6) × 6 = 288 cm

Now, When two cubes are place side by side so, the breadth and height of resulting cuboids will be same as cube but length will change

∴ Length of cuboid = 6 + 6 = 12 cm

So, length = 12 cm, breadth = 6 cm and height = 6 cm

∴ Total surface area of cuboids = 2 × (12 × 6 + 6 × 6 + 6 × 12) = 360 cm square

Now, the ratio of lateral surface area three cubes placed together to total surface area of all two cube placed together

∴ Required ratio = 288 ∶ 360 = 4 ∶ 5

Hence, option (2) is correct

Given:

The volume of each cube is 216 cm cube

Formula Used:

Side = ∛Volume of cube

Surface area of cube = 6 × (side) ²

Total Surface area of cuboids = 2 × (lb + bh + hl)

Required percentage = (First quantity/second quantity) × 100

Calculation:

The volume of cube is 216 cm cube

∴ Side of cube = ∛ 216 = 6 cm

Now, surface area of cube = 6 × (6) ² = 216 cm square

When three cubes are place side by side so, the breadth and height of resulting cuboids will be same as cube but length will change

∴ Length of cuboid = 6 + 6 + 6 = 18 cm

So, length = 18 cm, breadth = 6 cm and height = 6 cm

∴ Total Surface area of cuboids = 2 × (18 × 6 + 6 × 6 + 6 × 18) = 504 cm square

Now, required percentage = (504/216) × 100 = 233\(\frac{1}{3}\)%

Hence, option (1) is correct

Given:

Radius of a cone = radius of a Cylinder = radius of a hemisphere

Height of the cylinder = 2 × radius of the hemisphere

Height of the cone = 2 × height of the cylinder

Formula used:

Volume of cone = (1/3)πr²h

Volume of cylinder = πr²h

Volume of hemisphere = (2/3)πr³

Calculation:

Let, Height of hemisphere or radius of the hemisphere be r

Radius of cone = radius of Cylinder = radius of hemisphere = r

Height of cylinder = 2r

Height of cone = 4r

The volume of cone ∶ Volume of cylinder ∶ Volume of the hemisphere

= (1/3)πr²h ∶ πr²h ∶ (2/3)πr³

= [(1/3) × r² × 4r] ∶ [r² × 2r] ∶ [(2/3) × r³]

= (4/3) ∶ 2 ∶ (2/3)

= 4 ∶ 6 ∶ 2

∴ The ratio of the volumes of the cone to cylinder to hemisphere is 4 ∶ 6 ∶ 2.

Given:

Base of prism is square

Side of square base = 7 cm

Height of prism = 10 cm

Diameter of cylinder = 4.2 cm

Radius of cylinder = 4.2/2 = 2.1 cm

Height of cylinder = 10 cm

Concept:

The volume of the remaining solid is the difference of the volume of the prism and the cylinder which is carved out of the prism.

Formula used:

Volume of Prism = (Area of base) × Height of prism

Volume of Cylinder = πr²h

Area of square = Side × side

Calculation:

∵ Volume of remaining solid = Volume of prism – Volume of cylinder

= (7 × 7 × 10) – [(22/7) × 2.1 × 2.1 × 10]

= 490 – 138.6

= 351.4 cm³

Given:

Radius of a cylinder(R) = 8 cm

Height of cylinder = 10 cm

Radius of cone(r) = 3 cm

Height of cone = 4 cm

Concept:

The surface area of the remaining solid will be the sum of curved surface area of the cylinder, curved surface area of the cones, and the remaining area of the two circular faces of the cylinder.

Formula used:

Curved Surface area cylinder = 2πrh

Curved Surface area Cone = πrl

Slant height of cone = √(r² + h²)

Surface area of remaining circular face of cylinder = π(R² – r²)

Where ‘R’ is outer radius

And ‘r’ is inner radius

Calculation:

Curved surface area of cylinder = 2πRh

= 2 × π × 8 × 10

= 160π cm²       ------(1)

Slant height of cone = √[(3)² + (4)²]

= √(25)

= 5cm

∴ CSA of cone on both faces = 2πrl

= 2 × 5 × 3 × π

= 30π cm²      ------(2)

Surface area of both faces of cylinder = 2π(R² – r²)

= 2π[(8)² – (3)²]

= 2π(64 – 9)

= 2π × 55

= 110π cm²       ------(3)

Adding (1), (2) and (3);

Surface area of total solid = (Curved surface area of cylinder) + (CSA of cone on both faces) + Surface area of both circular faces of cylinder)

= 160π cm² + 30π cm² + 110π cm²

= 300π cm²

Given:

Radius of ball = 9 cm

Radius of cylinder = 6 mm = 0.6 cm

Height of cylinder = 1 cm

Concept:

The volume of the spherical ball will be equal to the total volume of all the cylinders that can be formed.

Formula used:

Volume of sphere = (4/3)πr³

Volume of cylinder = πr²h

Calculation:

Let, ‘n’  be the number of cylinders that can be formed;

∴ Volume of spherical ball = n × Volume of one cylinder.

⇒ (4/3)πr³ = n × πr²h

⇒ (4/3) × (22/7) × 9 × 9 × 9 = n × (22/7) × 0.6 × 0.6 × 1

⇒ n = 2700

Given:

Upper radius(R) = 33 cm

Lower radius(r) = 21 cm

Height of bucket = 42 cm

Concept:

Directly use the formula of volume of frustum and calculate the volume of the figure.

Formula used:

Volume of frustum = (1/3)πh(R² + r² + Rr)

Calculation:

Volume of frustum = (1/3)πh(R² + r² + Rr)

⇒ (1/3) × (22/7) × 42 × [(33)² + (21)² + (33 × 21)]

⇒ (1/3) × (22/7) × 42 × (1089 + 441 + 693)

⇒ 44 × 2223

⇒ 97812 cm³

∴ The volume of water it can hold is 97812 cm3.

Given: 

Total surface area of cube 864 cm²

Formula used: 

If one side of the cube be 'a' then,

Total surface area of the cube = 6a²

Volume of cube = a³

Calculation:

Accordingly,

6a² = 864

a²= 144

a = 12

Volume of cube = a³ = 12³

⇒  1728 cm³

∴ The volume of the cube is 1728 cm³.

Given:

Ratio between the length and the breadth of a rectangle = 17 : 13

Breadth is 5 cm less than the length

Formula used:

Perimeter = 2(Length + Breadth)

Calculation:

Let the length and breadth of rectangular lawn be 17a and 13a

According to the question  

⇒ 17a – 13a = 5

⇒ 4a = 5

⇒ a = 5/4 cm

Length = 17 × 5/4 = 21.25

Breadth = 13 × 5/4 = 16.25

Perimeter = 2(21.25 + 16.25) = 75 cm

∴ The perimeter of the rectangle is 75 cm

Radius of circle(r) = Breadth of rectangle(b)

The ratio of areas of circle and rectangle is 33 ∶ 16

⇒ (π × r²)/(l × b) = 33/16

l → length of rectangle

⇒ (π × b²)/lb = 33/16

⇒ 22b/7l = 33/16

⇒ l/b = 32/21

∴ Let l = 32x and b = 21x

∴ Required percentage = {(32x – 21x)/21x} × 100 = 52.38% ≈ 52.4%  

∴ The length of rectangle is 52.4% greater the breadth of the rectangle

Given:

The ratio of Sides is 11 ∶ 9

The difference of sides is 12 cm

Formula used

Area of rectangle = Length × Breadth

Calculation:

Let the ratio be x

∴ Length = 11x and Breadth = 9x

And the difference of sides is 12 cm

⇒ 12 = 11x – 9x

⇒ 2x = 12 cm

⇒ x = 6 cm

∴ Length = 11x = 11 × 6 = 66 cm and Breadth = 9x = 9 × 6 = 54 cm

Now, Area of rectangle = 11 × 9 × 6 × 6

⇒ (100 – 1) × 36

⇒ 3600 – 36 = 3564 square cm

∴ Area of rectangle is 3564 square cm

Measurement: It is defined as the description of data in terms of numbers. More precisely, measurement is defined as the assignment of numerals to objects or events according to rules.

3-dimensional body: ​A body that has length, breadth, and height. For example, glass, ball, and cylinder are three-dimensional objects as it has length, breadth, and height.

• A 3-dimensional body occupies space and hence, its volume can also be measured.
• Distance is measured between two points.
• Weigh is the measurement that signifies how heavy a body is.

Hence, we conclude that the extent of space occupied by a 3D body is its volume measure.

Given:

Length of diagonal of rectangle is 25 cm

And one of its sides is 24 cm

Formula used/Concept Used:

Length of diagonal of rectangle = √(length² + Breadth²)

Perimeter of rectangle = 2 × (Length + Breadth)

Calculation:

Let the other side of rectangle is x

∴ 25 = √24² + x²

Squaring both sides

⇒ x² = 49

⇒ x = 7 cm

Now, Perimeter of rectangle = 2 × (24 + 7) = 2 × 31 = 62 cm

∴ Perimeter of rectangle is 62 cm

Given

Length = (7/10) × breadth

Concept

Area of rectangle = length × breadth

Perimeter of rectangle = 2(length + breadth)

Calculation

Let the breadth of a rectangle is x cm

So, length = (7/10)x

Area of rectangle = 1750cm²

(7/10)x × x = 1750

⇒ x² = (1750) × (10)/7

⇒ x² = 2500

⇒ x = 50 cm

Perimeter of rectangle = 2[(7/10) × 50 + 50]

⇒ 2(85)

⇒ 170 cm

∴ The perimeter of rectangle is 170 cm 

Given:

Perimeter of square = 20 cm

Width of rectangle = Side of square 

Length of rectangle = 2 × Width of rectangle 

Formula used:

Perimeter of square = 4 × Side of square

Area of rectangle = Length × Width 

Calculations:
Let the width of rectangle be x

Side of square = x

Perimeter of square = 4x

⇒ 4x = 20

⇒ x = 5 cm

Length of rectangle = 2 × Width of rectangle 

Length of rectangle = 2 × 5

⇒ 10 cm

Area of rectangle = 5 × 10

⇒ 50 cm²

∴ The area of the rectangle is 50 cm2

Given:

Side of square = 39 cm

Length of rectangle = 2 × side of square

Concepts used:

Area of square = (Side)²

Area of rectangle = Length × Width

Calculation:

Area of square = (Side)² 

⇒ (39)² = 1521 cm².

Length of rectangle = 2 × side of square 

⇒ 2 × 39 cm

⇒ 78 cm.

Area of rectangle = Length × Width

As area of rectangle is to be kept equal to area of square,

⇒ 1521 = 78 × width

⇒ Width = 1521/78 

⇒ Width = 19.5 cm.

∴ The width of the rectangle is equal to 19.5 cm. 

Formula Used:

Area of square = Side × Side

Calculation:

Case:1

Let assume that side of a square = x unit

Area of square = Side × Side

⇒ Area of square  = x × x = x² square units   

Case: 2

⇒ Side of square is doubled , so side = 2x units

Area of square = Side × Side

⇒ Area of square  = 2x × 2x = 4x² square units

Ratio

⇒ Area of square 1/ Area of square 2 = x²/4x²

∴ Area of  square 1 ∶ Area of square 2 = 1 ∶ 4

The correct option is 2 i.e. 1 ∶ 4 .

Given-   

Height of prism = 10 m

Each Side of polygon = 2 m

Each interior angle = 120° 

Concept Used- 

Volume of Prism = Area of Base × Height 

Area of Hexagon = 3√3/2 × Side²

Each External Angle of a polygon = 360/n   [where n = number of sides of the polygon]

Calculation-

Exterior Angle = 180° - 120° 

⇒ 60° 

n = 360/60

⇒ 6

Hence the polygon is a regular hexagon.

Area of Hexagon = 3√3/2 × 2²

⇒ 6√3 m²

∴ Volume of Prism = 6√3 × 10     ∵ √3 = 1.7(given)

⇒ 102 m³

Given :-

Height of cone = 24 cm 

Volume of cone = 1232 cm2

Radius of cone = Radius of hemisphere

Concept :-

Volume of cone = (1/3)πr2h

Total surface area of hemisphere = 3πr2

Calculation :-

⇒ 1232 = (1/3) × (22/7) × r2 × 24

⇒ r2 = (1232 × 3 × 7)/(22 × 24)

⇒ r2 = 49

⇒ r = 7 

Total surface area of hemisphere = 3πr2

⇒ Total surface area of hemisphere = 3 × (22/7) × (7)2

⇒ Total surface area of hemisphere = 462 cm2

∴ Total surface area of hemisphere is 462 cm2

Given:

The sides of the cuboid are 44 cm, 32 cm, and 36 cm

The radius of the sphere is 12 cm

Concept Used:

The volume of a cuboid of sides l, b and h = l × b × h

The volume of the sphere of radius r = (4/3)πr³

Calculation:

The volume of the metallic cuboid is (44 × 32 × 36) cm³

The volume of the sphere is (4/3) × π × 12³

Let, the total number of such sphere is n

Accordingly,

44 × 32 × 36 = n × (4/3) × π × 12³

⇒ 44 × 32 × 36 = n × (4/3) × (22/7) × 12 × 12 × 12

⇒ n = 44 × 32 × 36 × (3/4) × (7/22) × (1/12) × (1/12) × (1/12)

⇒ n = 7

∴ Such 7 spheres can be made by given metallic cuboid.

GIVEN:

each side of a square is increased by 25%

FORMULA USED:

Area of square = (Side)² sq. unit

CALCULATION:

Let the side of the square be a unit

⇒ Area = a² sq. unit

side of a square is increased by 25%

⇒ New  side = 125a/100 

⇒ 5a/4

⇒ New area = (5a/4)² 

⇒ 25a²/16

⇒ Increased in area = (25a²/16 - a²)

⇒ 9a²/16

⇒ % increase in area = (9a²/16 × 1/a² × 100)

⇒ 56.25%

∴ % increase in area = 56.25%

Given :

Radius of base = 9 cm

Height of cone = 14 cm

Formula used :

Volume of cone = (1/3) × (22/7) × r² ×  h

Calculation :

Volume of cone = (1/3) × (22/7) × 9² ×  14

⇒ Volume of cone = 22 × 3 × 9 × 2

∴ Volume of cone is 1188 cm³

GIVEN:

Length and Breadth of the grassy plot are 110 m and 65 m and the width of the path be 2.5 m

FORMULA USED:

Area of the rectangular plot = (l × b) sq. unit

CALCULATION:

Area of the rectangular plot = (l × b) sq. unit

⇒ (110 × 65)

⇒ 7150 m²

Area of the plot excluding the path = (110 - 5) × (65 - 5)

⇒ 105 × 60 

⇒ 6300 m²

Area of the path = (Area of the rectangular plot - Area of the plot excluding the path)

⇒ 7150 - 6300

⇒ 850 m²

Cost of graveling the path = (850 × 80/100)

∴ Cost of graveling the path Rs680