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Given:

In ΔABC, AB = 8 cm and AC = 12 cm.

Semi-perimeter of ΔABC = 15 cm

Concepts used:

Semi-perimeter of triangle = Sum of three sides of a triangle/2

Calculation:

Semi-perimeter of triangle = Sum of three sides of a triangle/2

⇒ Semi-perimeter of ΔABC = (AB + BC + AC)/2

⇒ Semi-perimeter of ΔABC = (8 + BC + 12) cm/2

⇒ 15 cm = (20 + BC) cm/2

⇒ 15 cm × 2 = 20 cm + BC

⇒ BC = 30 cm – 20 cm

⇒ BC = 10 cm.

∴ The length of side BC is 10 cm.

Given:

The difference between the areas of a rectangle and square = 35 cm

Formula used:

Area of rectangle = length × breadth

Area of square = (side)²

Calculation:

Let the rectangle length and breadth be 'l' and 'b' respectively and the side of the square be 'a'

The length of a rectangle(l) = [(100 + 50)/100] × a

⇒ [150/100] × a

⇒ 1.5a

The breadth of a rectangle(b) = [(100 – 10)/100] × a

⇒ [90/100] × a

⇒ 0.9a

The difference between the areas of a rectangle and square = 35 cm

⇒ length × breadth – (side)2 = 35

⇒ 1.5a × 0.9a – a² = 35

⇒ 1.35 × a2 – a² = 35

⇒ 0.35 × a² = 35

⇒ a2 = 35/0.35

⇒ a2 = 100

⇒ a = 10 cm

The length of a rectangle = 1.5a

⇒ 1.5 × 10

⇒ 15 cm

The breadth of a rectangle = 0.9a

⇒ 0.9 × 10

⇒ 9 cm

The area of a rectangle = 15 × 9

⇒ 135 cm² 

∴ The area of a rectangle is 135 cm2. 

Given:

The area of the rectangle = 1764 cm2

Formula used:

Area of rectangle = length × breadth

Calculation:

Let the breadth of the rectangle be x then its length is 4x

Area = length × breadth

⇒ 4x × x = 1764

⇒ x² = 441

⇒ x = 21

Length of the rectangle = 4 × 21 = 84 cm

∴ The length of the rectangle is 84 cm

Given:

Breadth (B) of rectangle = Length (L) of rectangle - 10 cm

Product of sides = 24 cm² 

Concepts used:

Product of sides = Length × Breadth

Calculation:

B = L – 10 cm

L × B = 24 cm² 

⇒ L × (L – 10) = 24 cm² 

⇒ L² – 10L = 24 cm² 

⇒ L2 – 10L - 24 = 0

⇒ L² – (12 – 2)L - 24 = 0

⇒ L² – 12L + 2L – 24 = 0

⇒ L(L – 12) + 2(L – 12) = 0

⇒ (L – 12) (L + 2) = 0

⇒ (L – 12) = 0 or (L + 2) = 0

⇒ L + 2 = 0

⇒ L = -2 cm

As length (L) cannot be negative,

⇒ L – 12 = 0 

⇒ L = 12 cm.

⇒ B = L – 10 cm

⇒ B = 12 – 10 cm

⇒ B = 2 cm.

∴ The value of the breadth of the rectangle is 2 cm.

Given:

The area of the circle = 154 sq.cm

The breadth of the rectangle = 10 cm 

Formula used:

Perimeter of a rectangle = 2(length + breadth)

Circumference of a circle = 2πr

Area of the circle = πr²  

Calculation:

Area of the circle = πr2  = 154 cm²

⇒ Radius of a circle = √(154/π)= 7cm

⇒ Circumference of a circle = 2πr = 2 × 22/7 × 7 = 44cm

⇒ Perimeter of a rectangle = 2(length + breadth)

⇒ 2(l + 10) = 44 

⇒ Length of a rectangle = 22 - 10 = 12 cm

∴ Area of a rectangle = length × breadth = 12 × 10 ⇒ 120 sq.cm

Given:

The perimeter of a rectangle = 14 cm

Diagonal = 5 cm

Formula Used:

Perimeter of rectangle = 2(Length + Breadth)

Area of rectangle = Length × Breadth

(Diagonal)² = (Length)² + (Breadth)²

Calculation:

Perimeter of rectangle = 2(Length + Breadth)

⇒ 14 = 2(Length + Breadth)

⇒ 7 = (Length + Breadth)      ----(1)

⇒ (5)2 = (Length)2 + (Breadth)2

⇒ 25 = (Length)2 + (Breadth)²      ----(2)

From equation (1) and equation (2)

25 = (Length)2 + (7 - Length)²

⇒ 25 = l² + 49 - 14 l + l²

⇒ (l - 3)(l - 4) = 0

⇒ l = 3 cm

⇒ l = 4 cm

If length is 3 cm then breadth is 4 cm.

If breadth is 4 cm then the length is 3 cm.

Area of rectangle = Length × Breadth

∴ Area of rectangle = 3 × 4 = 12 cm²

The correct option is 2 i.e. 12 cm2

Given:

Length of rectangular cardboard = 11 m

Breadth of rectangular cardboard = 3 m

Formula Used;

Volume of Cylinder = πr²h , where r is radius of cylinder and h is height of cylinder.

Concept Used;

When a rectangular sheet is joined breadth to breadth, then the figure formed will be cylinder.

Length of Cardboard = Perimeter of base of cylinder

Breadth of cardboard = Height of cylinder

Calculation:

Using concepts,

We know

Perimeter of base of cylinder (2πr) = 11 m

⇒ 2 × (22/7) × r = 11

⇒ r = 7/4 m

Height of cylinder = 3 m

Volume of cylinder = (22/7) × (7/4)² × 3

⇒ V = 28.875 m³

∴ The Volume of the figure formed is 28.875 m³

Given:

Radius of First circle = p cm

Radius of Second circle = (p + 2) cm

Difference between areas of both the circles = 44 cm²

Formula Used:

Area of circle = π(Radius)²

(a + b)² = a² + b² + 2ab

Calculation:

Area of First circle = π(p cm)²

⇒ πp² cm²

Area of Second circle = π{(p + 2) cm}²

⇒ π(p² + 4p + 4) cm²

Difference between area of both the circles = π(p2 + 4p + 4) – πp2 

⇒ π(p2 + 4p + 4 – p²) = 44

⇒ 4π(p + 1) = 44

⇒ 4 × (22/7)(p + 1) = 44

⇒ p + 1 = (44 × 7)/88

⇒ p + 1 = 7/2

⇒ p = 7/2 – 1

⇒ p = 5/2

⇒ p = 2.5

∴ The value of p is 2.5

Given:

Height of cylinder, h = 24 m

Diameter of cylinder, d = 126 m.

Slant height of cone, l = 80 m.

Diameter of the cone, d = 126 m.

Breadth of canvas = 8 m.

Formula used:

Curved surface area of cylinder = 2πrh.

Curved surface area of cone = πrl

Calculation:

Diameter of cylinder, d = 126 m.

Radius of cylinder, r = 63 m.

Total area of tent = Curved surface area of cylinder + Curved surface area of cone 

⇒ 2πrh + πrl 

⇒ [2 × (22/7) × 63 × 24] + (22/7) × 63 × 80 

⇒ 9504 + 15840 

⇒ 25344 m

Length of canvas = (Total area of tent)/(Breadth of canvas)

⇒ 25344/8

⇒ 3168 m

∴The length of the canvas required to make the tent is 3168 m.

Given:

Ratio of the radius of two circle = 1 : 4

Formula used:

Area of circle = πr²

Calculation:

Let the radius of two circles be x and 4x

Area of the first circle = 22/7 × x × x

⇒ 22/7 × x²

Area of the second circle = 22/7 × 4x × 4x

⇒ 22/7 × 16x2

Ratio = 22/7 × x² : 22/7 × 16x²

⇒ x² : 16x²

⇒ 1 : 16

∴ Ratio of their area is 1 : 16

Given :-

Base of prism is square 

Height of prism = 10 cm

Volume of prism = 160 cm³

Concept :- 

Concept of prism is based on cylinder 

Total surface area of prism = Curved/lateral surface area + (2 × Base area)

Volume of prism = Base area × height 

Curved surface area of prism = Base perimeter × Height

Area of square = side²

Perimeter of square = 4 × side

Calculation :-

⇒ 160 = Base area × 10

⇒ Base area = (160/10)

⇒ Base area = 16 cm²

As base of prism is square so,

⇒ 16 = side²

⇒ Side = √16 = 4 cm 

⇒ Perimeter of base = 4 × 4 = 16 cm

Now 

⇒ Curved surface area = 16 × 10 = 160 cm²

⇒ Total surface area = 160 + (2 × 16)

⇒ Total surface area = 160 + 32 = 192 cm²

∴ Total surface area is 192 cm²

Given:

The radius of the cylinder = 14 cm

Height of the cylinder = 32 cm

Formula used:

Volume of cylinder = πr²h

Volume of cube = a³

Calculation:

According to the question,

Volume of the cylinder = πr²h

⇒ \(\frac{{22}}{7} × 14 × 14 × 32\)

⇒ 22 × 2 × 14 × 32

⇒ 19712

Side of cube = 4

No. of cubes = 19712/4³

⇒ No. of cubes = 19712/64

⇒ No. of cubes = 308

∴ The number of cubes that can be made is 308.

Given:

The sides of three cubes = 15 cm, 20 cm, and 25 cm respectively

Concept:

When the number of solids reshapes in on solid then the volume of the solid is the same

Formula used:

The volume of the cube = a³

(Where a = The side of the cube)

Calculation:

Let us assume the side of the single cube be X

⇒ The volume of the first cube = 15³ = 3375 cm³

⇒ The volume of the second cube = 20³ = 8000 cm³

⇒ The volume of the third cube = 25³ = 15625 cm³

⇒ The total volume of all three cubes = The volume of the reshaped single cube

⇒ X³ = 3375 + 8000 + 15625

⇒ X³ = 27000

⇒ X = ∛27000 = 30 cm

∴ The required result will be 30 cm.

Surface area of cube = 6 × a², where a is side of the cube

Volume of cube = a³

Volume of cube with side 1 cm = 1³ = 1 cm³

Volume of cube with side 6 cm = 6³ = 216 cm³

Volume of cube with side 8 cm = 8³ = 512 cm³

Total Volume = 1 + 216 + 512 = 729 cm³

Side of new cube = a cm

Volume of new cube = a³ cm³

a³ = 729

⇒ a = 9

Side of cube = 9 cm

Surface Area = 6 × a² = 6 × 9² = 486 cm²

∴ Surface area of the new formed cube 486 cm²

Given:

Diameter of wheel = 3 m

Total distance = 6.6 km = 6600 m

Circumference of circle = 2π r where r = radius of the circle

Diameter of circle = 2 × r

Concept used:

A wheel cover its length of circumference while making 1 revolution

Radius of wheel = 3/2 = 1.5 m

Total number of revolution = Total distance / circumference of wheel

⇒ n = 6600/2πr = 6600 / (2π × 1.5) = 6600 × 7/(2 × 22 × 1.5) = 700

GIVEN:

area of the parallelogram is 72cm2

FORMULA USED:

Area of parallelogram = (Base × Height) sq. unit

CALCULATION:

Let the height of the parallelogram be x cm  and the Base be 2x cm

⇒ Area of parallelogram = (Base × Height) sq. unit

⇒ 72 = x × 2x

⇒ 72 = 2x²

⇒ x² = 36

⇒ x = 6

∴ Height of the parallelogram is 6 cm

Given:

Area of the parallelogram = 24 cm2

Base = 6 cm

Formula Used:

Area of parallelogram = Base × Height

Calculation:

Area of parallelogram = Base × Height

⇒ 24 cm² = 6 cm × x

⇒ x = 4 cm

∴ Height of the parallelogram is 4 cm.

The correct option is 1 i.e. 4 cm.

Given:

Area of parallelogram = 22 cm² 

Base = (x + 5) cm

Height is 9 cm less than base

Formula used:

Area of parallelogram = Base × Height

Calculation:

Height of parallelogram = (x + 5) – 9 = x – 4 cm

Area of parallelogram = Base × Height

⇒ Area = (x + 5) × (x – 4)

⇒ 22 = x² + x – 20

⇒ x² + x – 42 = 0

⇒ x² + 7x – 6x – 42 = 0

⇒ x × (x + 7) – 6 × (x + 7) = 0

⇒ (x + 7) × (x – 6) = 0

⇒ x = –7 and x = 6

Height = x – 4 = 6 – 4

∴ Height of parallelogram is 2 cm

Given:

The surface area of the sphere = 1386 cm²

Formula used:

Surface area of a sphere = 4πr²

Volume of a sphere = (4/3)πr³

Calculations:

Surface area of a sphere = 4πr2

⇒ 4πr² = 1386

⇒ 4 × (22/7) × r² = 1386

⇒ r = 21/2 cm

Volume of a sphere = (4/3)πr³

⇒ (4/3) × (22/7) × (21/2)³ 

⇒ 4851 cm³

∴ The volume of the sphere is 4851 cm³.

Given:

Area of circle = 154 m²

Formula used:

Area of circle = πr²

Perimeter of circle = 2πr

Calculation:

πr² = 154 m²

⇒ (22/7) × r² = 154 m²

⇒ r² = 7 × 7

⇒ r = 7 m

Perimeter of circle = 2 × (22/7) × 7 m

⇒ 44 m

∴ The perimeter of circle is 44 m

Given:

Circumference of the base of cylinder = 66 cm

Height of cylinder = 40 cm

Formula Used:

Circumference of a circle = 2πr

Where r = Radius of circle

Volume of cylinder = πr²h

Where h = height of the cylinder

Calculation:

Let the radius of the base of the cylinder be r

2πr = 66 cm

⇒ 2 × 22/7 × r = 66 cm

⇒ r = (3 × 7)/2 cm

⇒ r = 21/2 cm

Volume of cylinder = π(21/2 cm)² × 40 cm

⇒ 22/7 × 441/4 × 40 cm³

⇒ 220 × 63 cm³

⇒ 13860 cm³

∴ The Volume of the cylinder is 13860 cm³

Given:

The circumference of a circle exceeds its diameter by 30 cm. 

Concept used:

Circumference and area of circle

Calculation:

As per the question,

⇒ 2 π r - 2 r = 30

⇒ 2r(π - 1) = 30

⇒ \(2r\left( {\frac{{22}}{7} - 1} \right) = 30\)

⇒ 2r = 14

⇒ r = 7 cm

Area of circle = π r²

Area = \(\frac{{22}}{7} \times 7 \times 7 = 154\) cm²

Given:

 circumference of a circular garden = 176 cm

Formula Used:

Circumference of circle = πd

Calculation:

Circumference of circular garden = Circumference of circle = 176 cm

⇒ 176 cm = πd

⇒ 176 cm × 7/22 = d

⇒ 56 cm = d

∴ The diameter of the circular garden is 56 cm.

The correct option is 1 i.e. 56 cm.

Given :

Side of the cube = a cm 

Area of base of pyramid = a² cm²

Height of the pyramid = a cm 

Formula used : 

Volume of the cube = a³ 

Volume of the pyramid = (1/3) × (Area of base × height)

Calculations :

Volume of the cube = a³

Volume of the pyramid = (1/3) × (a² × a) = (1/3)a³

Ratio of volume of cube to volume of pyramid = a³ : (1/3)a³

⇒ 3 : 1 

∴ The ratio of the volume is 3 : 1

Given:

Cube of edge 18 cm, 19 cm, and 21 cm.

The radius of cone = 49/11 cm

Formula used:

The volume of cube = side³

The volume of cone = (1/3)πr²h

Calculation:

The volume of newly shaped cube = sum of the volume of all cube

Let the side of three cubes be a₁, a₂, a₃.

The volume of all cube

⇒ (a₁)³ + (a₂)³ + (a₃)³

⇒ (18)³ + (19)³ + (21)³

⇒ 21952

The volume of cone = The volume of all cube

⇒ (1/3)πr²h = 21952

⇒ h = (21952 × 3 × 7 × 11 × 11)/(22 × 49 × 49)

⇒ h = 1056 cm

⇒ h = 10.56 m

Let the side of square = x and the radius of circle = r;

∵ The circle and the square have the same perimeter;

∴ 4x = 2πr

⇒ x = πr/2

Area of circle = πr2

Area of square = x2 = (πr/2)2 = π²r2/4

⇒ Area of square =22/28 × Area of circle

∴ Required ratio = 11 : 14

Given:

Radius of the cone = 7cm

Height of the cone = 21 cm 

Formula Used:

The volume of the cone = (πr²h)/3

Calculation:

The volume of the cone = (π × 7² × 21)/3 = 343π 

The volume of cone = 343 × (22/7) = 1078 cubic cm

∴ The volume of the cone is 1078 cubic cm

Given:

The capacity of the room = 1680 m³

The area of the floor = 112 m²

Formula Used:

Volume of the cuboid = l × b × h

Area of the floor = l × b

Calculation:

The height of the room = Volume of the room/Area of the floor =( l × b × h)/(l × b)

Height of the  room = 1680/112 = 15 m

∴ The height of the room is 15 m

Given:

Area of the rectangular plot is 2000 m²

Calculation:

Let breath of the rectangle be x m

So, length = 5x m

5x m × x m = 2000 m²

⇒ 5x² m² = 2000 m²

⇒ x² = 400

⇒ x = 20

⇒ Breath = 20 m

∴ The breadth of the rectangular plot is 20 m

Given:

difference between areas is 44 sq. cm 

radii = p cm and (p+2) cm

Formula Used:

Area of Circle = π.r²

Calculation:

π.R² - π.r² = 44

⇒π(R² - r²) = 44

⇒22/7{(R + r)(R - r)} = 44

⇒{(p+2+p)(p+2-p)} = 14

⇒(2p + 2)2 = 14 

⇒2p+ 2 = 7

⇒2p = 5

∴ The value of p is 2.5

Given:

The ratio of the circumference of two circles = 5 ∶ 8

Formula used:

Circumference of a circle = 2 π R

Area of a circle = π R²

Calculations:

Let the radius of two circles be r₁ and r₂

The ratio of the circumference of two circles = (2 × π × r₁) ∶ (2 × π × r₂)

⇒ r₁ ∶ r₂ = 5 ∶ 8

The radius of two circles are 5x and 8x

Ratio of corresponding areas = (π × r₁²) ∶ (π × r₂²)

⇒ r₁² ∶ r₂² = (5x)² ∶ (8x)²

⇒ 25x² ∶ 64x²

⇒ 25 ∶ 64

∴ The ratio of their corresponding areas is 25 ∶ 64

Given:

Length of the roller is 1.2 m. Its internal radius is 24 cm and thickness is 15 cm.

1 cm3 of iron has 8 g mass

Concept Used:

If R be the external radius and r be the internal radius and h be the height of a cylinder then the volume of the material of the cylinder is π(R² - r²)h

Calculation:

Length of the roller (h) = 1.2 m = 120 cm

The internal radius of the cylinder (r) = 24 cm

The thickness of the cylinder is 15 cm

External radius of the cylinder (R) = (24 + 15) = 39 cm

The volume of the cylinder is π(39² - 24²) × 120

⇒ 113400π cm³

Mass of 1 cm³ iron is 8 g

Mass of 113400π cm³ iron is (113400π × 8) g

⇒ 907200π g

⇒ 907.2π kg         [∵ 1 kg = 1000 g]

∴ The mass of the roller is 907.2π kg.

Given:

Area of trapezium = 350 m²

Two parallel side are 25 m and 10 m

Formula used:

Area of trapezium = (1/2) × (Sum of Parallel sides) × (Distance between Parallel sides)

Calculation:

Let d is the distance between 2 parallel sides

Area of trapezium = (1/2) × (Sum of Parallel sides) × (Distance between Parallel sides)

⇒ 350 = (1/2) × (25 + 10) × d

⇒ 700 = 35 × d

⇒ d = 20 m

∴ Distance between two parallel sides is 20 m.

Given:

L₁ = 17 cm

L₂ = 15 cm

Height (H) = 6 cm

Concept used:

Area of trapezium = (1/2) × (Sum of parallel side) × Height 

1 cm = (1/100) m 

Calculation:

Let two parallel side be L1 and L2

Area of trapezium = (1/2) × (17 + 15) × 6

⇒ 96 cm²

⇒ 96/10000 m² = 0.0096 m²

∴ The area of the trapezium is 0.0096 m².

Given:

One of the parallel side is 17 cm

Distance between parallel sides is 10 cm

Formula Used:

Area = Sum of parallel sides × height/2

Calculation:

Let the other side be x

⇒ (17 + x) × 10/2 = 120

⇒ 17 + x = 24

⇒ x = 7

∴ The Required ratio is 7 ∶ 17

Given:

Side of a rectangular field  = 48 m

The diagonal of a rectangular field = 73 m

Concept used:

By using pythagoras theorem

Diagonal2 = length2 + breadth2

Formula required: 

Area of the rectangle = length × breadth

Calculations:

73² = 48² + length²

⇒ Length² = 73² - 48²

⇒ Length² = 5329 - 2304 = 3025

⇒ Length = √3025

⇒ Length = 55

Area of rectangular field = 55 × 48

⇒ 2640 m²

∴ The area of rectangular field is 2640 m²

Given:

The side of the triangle is 7.5 cm, 4 cm and 9 cm respectively and circum radius is 1.5 cm

Formula used:

Area of triangle = (Product of sides)/ 4 × circum radius

Calculation:

By using the formula

∴ Area of triangle = (7.5 × 4 × 9)/(4 × 1.5) = 45 square cm

Hence, option (2) is correct

Given:

The sum of all sides of the triangle is 48 m and in radius is 7.5 m

Formula used:

Area of the triangle = (sum of all sides × r)/2

Where r is the radius of the incircle

Calculation:

By using the given formula

∴ Area = (48 × 7.5)/2 = 180 square m

Hence, option (1) is correct

Given:

The circum radius of equilateral triangle is 2.5√3

Formula used:

Circum radius of equilateral triangle = side/√3

Calculation:

The circum radius of equilateral triangle is 2.5√3

∴ Side of equilateral triangle = 2.5√3 × √3 = 7.5 cm

Hence, option (1) is correct

Given:

Radius of cylindrical part = 0.7 cm

Height of cylindrical part = 4 cm

Volume of Bullet = 7.7 cm³

Concept:

The radius of the conical part will be same as the radius of the cylindrical part because it is mounted on it. The mass of the bullet will be the sum of the mass of the cylindrical and conical part.

Formula used:

Volume of cone = (1/3)πr²h

Volume of cylinder = πr²h

Calculation:

Let, the height of the conical part = ‘a’ cm

∵ Volume of Cylindrical part = πr²h

= (22/7) × 0.7 × 0.7 × 4

= 6.16 cm³

Volume of conical part = (1/3)πr²h

= (1/3) × (22/7) × 0.7 × 0.7 × a

= (10.78 × a)/21 cm³

∵ Volume of bullet = Volume of Cylindrical part + Volume of conical part

⇒ 7.7 = 6.16 + (10.78 × a)/21

⇒ 7.7 × 21 = (6.16 × 21) + (10.78 × a)

⇒ 161.7 = 129.36 + 10.78a

⇒ 161.7 – 129.36 = 10.78a

⇒ 32.34 = 10.78a

⇒ a = 32.34/10.78

⇒ a = 3 cm

Given:

Outer diameter = 12 cm

Outer radius(R) = 12/2 = 6 cm

Thickness = 1 cm

∴ Inner radius(r) = Outer radius – thickness

= (6 – 1) cm = 5 cm

Formula used:

Volume of a Hemispherical bowl = (2/3)π(R³ – r³)

Calculation:

∵ Volume of a Hemispherical bowl = (2/3)π(R³ – r³)

= (2/3) × π × [(6)³ – (5)³]

= (2/3) × (22/7) × (216 – 125)

= (2/3) × (22/7) × 91

= (182/3) × (22/7)

= 190.6 cm³

≈ 191 cm³

Given - 

base of the pyramid is an equilateral triangle of side 10 m, height of pyramid = 40√3 m

Formula used - 

volume of pyramid = (1/3) × area of base × height

area of equilateral triangle = (√3/4) × side²

Solution - 

Let the volume of pyramid be V m³.

⇒ Area of Equilateral triangle = (√3/4) × 10² = 25√3 m²

⇒ V = (1/3) × 25√3 × 40√3

⇒ V = 1,000 m³

∴ Volume = 1,000 m³.

Given:

Side of Base triangle = 8 cm

Volume of pyramid = 96 cm³

Concept:

Using the formula of volume of the pyramid, calculate the height.

Formula used:

Volume of pyramid = (1/3) × (Area of base) × height

Area of equilateral triangle = (√3/4) × (side)²

Calculation:

Let, the Height of the pyramid = ‘x’

Area of equilateral triangular base = (√3/4) × 8 × 8

= 16√3 cm²

∴ Volume of pyramid = (1/3) × (Area of base) × height

⇒ 96 = (1/3) × 16√3 × x

⇒ x = (96 × 3)/16√3

⇒ x = 18/√3 cm

⇒ x = 6√3 cm

Given:

Ratio between CSA = 5 ∶ 39

Ratio between diameters = 1 ∶ 3

Formula used:

CSA of cone = πrl, where l is slant height and r is the radius of the base.

Calculation:

Let the l₁ and l₂ be the slant height of two cones.

Curved surface area of cone 1, CSA₁ = πR₁l₁

Curved surface area of cone 2, CSA₂ = πR₂l₂

According to the question,

CSA₁/CSA₂ = 5/39

⇒ (πR₁l₁)/( πR₂l₂) = 5/39

⇒ R₁/R₂ × l₁/l₂ = 5/39

⇒ 1/3 × l₁/l₂ = 5/39

⇒ l₁/l₂ = 5/13

∴ The ratio between their slant height is 5 ∶ 13.

Explanation: 

(i) Congruent figures have the same size, the same angles, the same sides and the same shape. They are Identical.

(ii) Congruent shapes are always similar, but similar shapes are usually not congruent one is bigger and one is smaller.

In congruent shapes the ratio of the corresponding sides is 1 : 1

Given-

Length of diagonal = 162 cm

Ratio of length, breadth and height of the cuboid = 8 : 4 : 1

Formula Used-

Diagonal of a cuboid = \(\sqrt{l^2 + b^2 + h^2}\)l2+b2+h2−−−−−−−−−−√l2+b2+h2

Calculation -

Let  the length, breadth and height of cuboid be  8x, 4x and  x

Now,

\(\sqrt{8x^2 + 4x^2 + x^2}\) = 162

⇒ \(\sqrt{81x^2}\) = 162

⇒ 9x = 162

⇒ x = 18 cm

∴ Height of cuboid = 1 × 18 = 18 cm

Given:

The length of the arms of a rhombus is 16 cm and 12 cm.

Concept Used:

If the length of the diagonals of a rhombus are d₁ and d₂ then the sides of the rhombus is (1/2) × √(d₁² + d₂²)

The perimeter of a rhombus is (4 × Length of the side of the rhombus)

Calculation:

Here d₁ = 16 cm

and d₂ = 12 cm

The length of the sides of the rhombus is (1/2) × √(12² + 16²)

⇒ (1/2) × √(144 + 256)

⇒ (1/2) × 20

⇒ 10

The perimeter of the rhombus is 4 × 10 = 40 cm

∴ The perimeter of the rhombus is 40 cm.

 Length of arms means the length of diagonals.

Given

The altitude of an equilateral triangle = 4√3

Concept

Altitude of an equilateral triangle = (√3/2) × side

Area of an equilateral triangle = (√3/4) × (side)²

Calculation

According to the question

4√3 = (√3/2) × side

⇒ Side = [(4√3) × 2]/√3

⇒ Side = 8 cm

Now, 

⇒ Area of an equilateral triangle = (√3/4) × 8²

⇒ Area of an equilateral triangle = 16√3 cm²

∴ Area of an equilateral triangle = 16√3 cm2