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Find the altitude of a right prism for which the area of the lateral surface is 143 sq.cm and the perimeter of the base is 13 cm.

Option 2 : 11 cm

**Given:**

Area of the lateral surface of Prism(L.S.A) = 143 sq.cm

Perimeter of the base, P = 13 cm

**Concept:**

Lateral surface area of a prism is the product of its base perimeter and altitude

**Formula used:**

Lateral surface area of Prism = Perimeter of base × Altitude (L.S.A = P × h)

**Calculation:**

L.S.A = P × h

⇒ h = L.S.A/P = 143/13 = 11 cm

∴ Altitude of the Right Prism = 11 cm

Option 1 : 54 cm^{3}

**Given:**

Hypotenuse of base triangle = 3√2 cm

Height of prism = 12 cm

**Formula used:**

In a right angled triangle;

(Hypotenuse)^{2} = (Base)^{2} + (Height)^{2}

Area of triangle = (1/2) × Base × Height

Volume of Prism = Area of base × Height

__ Calculation__:

In Isoceles triangle; two sides are equal.

Let the equal sides = a cm

(3√2)^{2} = a^{2} + a^{2}

⇒ 18 = 2a^{2}

⇒ a^{2} = 9 cm

⇒ a = 3 cm

Area of triangular base = (1/2) × 3 × 3

⇒ 9/2 cm

Volume of Prism = 12 × (9/2) cm^{3}

⇒ 54 cm^{3}

**∴ The volume of prism is 54 cm ^{3}.**

Option 4 : 1260

Given,

Height of the triangular prism is h = 21 cm

The ratio of sides of triangular prism is = 8x : 15x : 17x

The total area of three lateral surfaces is = 840 cm^{2}

Perimeter of the triangle × height = 840

⇒ (8x + 15x + 17x) × 21 = 840

⇒ 40x = 840/21

⇒ x = 40/40

⇒ x = 1

Sides of the triangular prism are 8, 15 and 17.

As we know,

⇒ s = (a + b + c)/2

⇒ s = (8 + 15 + 17)/2

⇒ s = 40/2 = 20

Area of the triangle = √[s (s – a) (s – b) (s – c)]

Area of the triangle = √[20 × (20 – 8) (20 – 15) (20 – 17)]

Area of the triangle = √[20 × 12 × 5 × 3]

Area of the triangle = 60

As we know,

Volume of the prime = Area of Base × height

Volume of triangular prism = 60 × 21 = 1260 cm^{3}

Option 2 : 28 cm^{2}

**Given:**

The volume of a prism is 308 cm3 and the height is 11 cm.

**Formula used:**

The volume of prism = area of its base × height

**Calculation:**

According to the question,

308 = area of its base × 11

Option 2 : 12

Since, volume of prism = Area of base × height

So, 288 = Area of base × 24

∴ Area of base = 12 cmOption 1 : 36 cm^{2}

Since the base surface area of all the pieces becomes 8 times of the base area of the original prism. Hence, the total number of pieces must be equal to 8.

Volume of each small piece = 96√3/8 = 12√3 cm^{3}

Base area × Height = 12√3

(√3/4) × 4^{2} × Height = 12√3

Height = 12√3/4√3 = 3 cm

Lateral surface area of each piece = Perimeter of base × Height

= (3 × 4) × 3

= 36 cmOption 1 : 649500 cm^{3}

**Given:**

The ratio of the side of the base and height of a hexagonal prism is 1 ∶ 2. The perimeter of the base of hexagonal prism is 300 cm

**Calculation:**

Perimeter of the base of hexagonal prism = 300 cm

⇒ Side of hexagon = 300/6 = 50 cm

Then,

⇒ (side of hexagon) : (height of hexagon) = 1 : 2

⇒ Height of hexagon = 2 × 50 = 100 cm

Area of hexagon = 6 × √3/4 × side^{2}

= 6 × √3/4 × 50 × 50

= 3750√3 sq. cm

Volume of hexagonal prism = Base area × Height

⇒ 3750√3 × 100

⇒ 375000 × √3

⇒ 649519 cm^{3 }

**∴ From the options, Volume of hexagonal prism is 649500 cm3**

The base of an even solid prism is a triangle whose sides are 6, 8 and 10 cm. The height of the prism is 10 cm. What will be the total surface area, lateral surface area and volume of the prism?

Option 4 : 288 sq cm, 240 sq cm, 240cubic cm

**Solution:**

**Given: **Lateral height (L) = 10 cm

Hypotenuse (a) = 10 cm

Base of triangle (b) = 8 cm

Height of triangle (h or c) = 6 cm

**Formula used:**** **Total surface area = 2 [ (1 / 2) (b) (h) ] + (a + b + c) × L

Lateral surface area = (a × L) + (b × L) + (c × L)

Volume = (1 / 2) (b) (h) × L

**Calculation:** Total surface area = 2 [ (1 / 2) (8) (6) ] + (8 + 10 + 6) × 10

⇒ 2 [4 × 6] + 24 × 10

⇒ 2 × 24 + 24 × 10

⇒ 48 + 240

⇒ 288

Lateral area = (10 × 10) + (8 × 10) + (6 × 10)

⇒ 100 + 80 + 60

⇒ 240

Volume = (1 / 2) (8) (6) × 10

⇒ 4 × 6 × 10

⇒ 240

**∴ Total surface area = 288 cm ^{2}**

**Lateral surface area = 240 cm ^{2}**

**Volume = 240 cm ^{3}**

Option 2 : 6

**Calculation:**

The triangular prism has 2 triangular faces and 3 rectangular faces and has 6 corners.

Option 1 : 7

**Given:**

Lateral surface area of prism is 252 sq. cm

Volume of the prism is 147√3 cm^{2}

**Formula used:**

Lateral surface area of prism = perimeter of base × height

Volume of the prism = Area of base × height

**Calculation:**

Let the side of the base of prism be a cm and height of the prism be h cm.

Lateral surface area of prism = perimeter of base × height

⇒ Lateral Surface Area of Prism = 3 × a × h = 252

⇒ ah = 252/3

⇒ ah = 84 ---- (1)

The volume of the prism = Area of base × height

\(\Rightarrow 147\sqrt 3 = \frac{{\sqrt 3 }}{4} \times {\left( a \right)^2} \times h\;\)

\(\Rightarrow 147\sqrt 3 = \frac{{\sqrt 3 }}{4} \times a \times ah\)

\(\Rightarrow 147\sqrt 3 = \frac{{\sqrt 3 }}{4} \times a \times 84\;\)

⇒ 147 = 21 × a

⇒ a = 7 cm

∴, the side of the base of the prism is 7 cm.

Option 4 : (48√3 + 192) cm^{2}

**Given:**

The base of a prism is a regular hexagon of side length 4 cm

The volume of the prism is 192√3 cm^{3}

**Calculation:**

Let the height of prism be h cm

Since, volume of prism = 192√3

So, area of base × height = 192√3

6 × √3/4 × 4^{2} × h = 192√3

So, h = 8 cm

Now, total surface area = 2 × area of base + lateral surface area

⇒ 2 × 6 × √3/4 × 4^{2 }+ 6 × 4 × 8

Option 1 : 180√3

Perimeter of base (equilateral triangle) = 3 × side = 18cm

⇒ Side of base = 6 cm

⇒ Area of base = side^{2} × (√3/4)

⇒ Area of base = (√3/4) × 6 × 6

⇒ Area of base = 9√3cm

⇒ Volume of prism (V) = area of base × height

∴ Volume of prism (V) = 9√3 × 20 = 180√3 cmThe base of a right prism is a right-angled triangle and the measure of the base of the right-angled triangle is 12 m and its height is 16 m and If the height of the prism is 9 m then what is the number of edges of the prism and its volume?

Option 1 : 9 and 864 m^{3}

**Given:**

The base and height of the right-angled triangle are 12 m and 16m respectively

**Formula used:**

Area of the right-angled triangle = ½ × base × height

The volume of the prism = Area of the base × Height of the prism

The number of the edges of the prism = The number of the sides of the base × 3

**Calculation:**

The number of the edges of the prism = The number of the sides of the base × 3

⇒ 3 × 3 = 9

Area of the base = ½ × 12 × 16 = 96 m2

The volume of the prism = 96 × 9 = 864 m3

**∴ The number of edges and its volume is 9 and 864 m3**

Option 3 : 1440 cm^{3}

**GIVEN:**

Lateral surface area of the prism is 240√3 cm^{2}

Height of prism = 5√3 cm

**CONCEPT:**

Value of ‘x’ can be found with help of LSA.

**FORMULA USED:**

Lateral surface area of the prism = Perimeter of the base × Height

Volume of the prism = Base area × Height

Area of hexagon = 6 × (√3/4) × (side)^{2}

**CALCULATION:**

Perimeter of the base of prism = 6 × x = 6x

Lateral surface area of the prism = 6x × 5√3 = 240√3

⇒ 6x = 48

⇒ x = 8 cm

Base area of the prism = 6 × (√3/4) × 8^{2} = 96√3 cm^{2}

⇒ Volume of the prism

⇒ 96√3 × 5√3

⇒ 1440 cm^{3}

∴ volume of prism is 1440 cm^{3}

Option 4 : 192√3 cm^{3}

Area of base of equilateral triangle = (√3/4) × 8 × 8 = 16√3

Volume = area of base × height

= 16√3 × 12 = 192√3 cmOption 4 : 2520

As we know

(Hypotenuse)^{2} = (Base)^{2} + (Perpendicular)^{2}

⇒ 29^{2} = 21^{2} + 20^{2}

⇒ 841 = 441 + 400

⇒ 841 = 841

Now we can say triangle is a right-angled triangle.

As we know,

Volume of prism = Area of base × height

⇒ (1/2) × 20 × 21 × h = 7560

⇒ h = (7560 × 2)/(20 × 21) = 36 cm

As we know

Lateral surface area of prism = Perimeter of base × Height

Lateral surface area of prism = (29 + 21 + 20) × 36 = 2520 cm^{2}

Option 1 : 2√3 cm

Let the side of the equilateral triangle be ‘x’ cm and the length of the prism be ‘l’ cm

∵ The triangular prism comprises of 3 equal faces,

⇒ Lateral surface area of prism = 3lx = 60√3 cm^{2}

⇒ lx = 20√3 cm^{2}

Now,

Volume of prism = Area of base × Length of prism

∵ Area of equilateral triangle = (√3/4) × (side)^{2} = (√3/4)x^{2} cm^{2}

⇒ Volume of prism = (√3/4)lx^{2} cm^{3}

Substituting for ‘lx’,

⇒ 30√3 = (√3/4) × (20√3)x

⇒ x = 30√3/15 = 2√3 cm

∴ Side of equilateral triangle = 2√3 cmThe height of the two prisms is 14 cm and 10 cm. The ratio of the area of bases of two square prisms is 7 ∶ 9 and the volume of the first prism is 882 cm^{3} then what is the total surface area of the second prism?

Option 1 : 522 cm2

Given:

The heights of the two prisms are 14 cm and 10 cm

The ratio of the area of bases of two square prisms are 7 ∶ 9

The volume of first prism = 882 cm3

Formula used:

The volume of the square prism = Area of the base × height

The total surface area of the square prism = 2 × Area of base + 4 × Area of lateral sides

Calculation:

Let the area of the base of the first and second prism be 7x and 9x

The volume of the first prism = Area of the base × height

⇒ 7x × 14 = 882

⇒ x = 9

So, Area of the base of the second prism = 9 × 9 = 81

So, The length of the side of the second prism = √81 = 9 cm

Now, Total surface area of the second prism = 2 × area of base + 4 × area of the lateral side

⇒ 2 × 92 + 4 × 9 × 10 = 522 cm2

**∴ The Total surface area of the second square prism is 522 cm2**

Option 3 : 1350

**Given :-**

The base of a right prism is a regular hexagon of side 5 cm

height is 12 √3 cm

**Concept :-**

Prism is a part of cylinder so,

Volume of prism = Base × Height

As base of prism is shan so

Volume of prism with base hexagonal = base area × height

Base area = Area of hexagonal is equal to area of 6 equilateral triangle = 6 × (√3/4) × side^{2}

**Calculation :-**

⇒ Base area = 6 × (√3/4) × 5^{2}

⇒ Base area = 150 × (√3/4)

⇒ Volume = 150 × (√3/4) × 12√3

⇒ Volume = (1800 × 3)/4

⇒ Volume = 1350 cm^{3}

**∴ Volume = 1350 cm ^{3}**

Option 4 : 1 : 1 : 1

**Concept used:**

Square-based prism is generally cube or cuboid in shape

**Calculation:**

When the prism is cut in three parts of equal heights then the volumes of all parts are equal

So, the ratio of the volume of the top, middle and the bottom part = 1: 1: 1

**∴ The required ratio is 1 ∶ 1 ∶ 1**