Question

# What is the length of the longest pole that can be fit in the room of dimensions 5m x 10m x 5m

- 12.25
- 11.95
- 12.85
- 11.85

Hint:

### Diagonal of cube= × side = a unit

## The correct answer is: 12.25

### The dimensions of the room are

Length = 5m

Breadth = 10m

Height = 5m

We know that

Length of the longest pole = length of diagonal =

By substituting the values

Length of the longest pole =

So we get

Length of the longest pole =

By addition

Length of the longest pole =

Therefore, the length of the longest pole that can be put in the room is 12.25m.

The correct option is a)12.25.

### Related Questions to study

### The square on the diagonal of a cube has an area of 192 cm^{2}. Find the T.S.A of the cube.

Length of diagonal on one face of cube = l

Length of diagonal of the cube = z

We know that,

Given z²=192 cm²

Squaring the equation of Z on both sides we get,

192 = s²×3

s² =192/3

s^{2 }= 64

s = 8 cm

Total surface area of cube = 6s²

TSA = 6×8² = 6×64 = 384 cm²

Hence side of given cube = S = 8 cm

and Total surface area of cube =TSA= 384 cm²

Therefore, the correct option is d) 384.

### The square on the diagonal of a cube has an area of 192 cm^{2}. Find the T.S.A of the cube.

Length of diagonal on one face of cube = l

Length of diagonal of the cube = z

We know that,

Given z²=192 cm²

Squaring the equation of Z on both sides we get,

192 = s²×3

s² =192/3

s^{2 }= 64

s = 8 cm

Total surface area of cube = 6s²

TSA = 6×8² = 6×64 = 384 cm²

Hence side of given cube = S = 8 cm

and Total surface area of cube =TSA= 384 cm²

Therefore, the correct option is d) 384.

### Two adjacent sides of the right-angle pf a right-angled triangle are 3 cm and 4 cm. This triangle is revolved around the side of length 3 cm. Find the volume of the cone thus formed. Find also the slant height of the cone (π =3.14).

Therefore by pythagorous theorem third side (slant height) will be

It is revolved about 3 cm side

Therefore, the cone formed will have dimensions as

Height, h = 3 cm

Radius, r = 4 cm

So, the volume of cone = (1/3)πr

^{2}h

= (1/3)(22/7)(4 x 4)(3)

= 22/7 x 16

= 352 / 7

= 50.28 cm

^{3}

Therefore, the correct option is c) 5 cm

### Two adjacent sides of the right-angle pf a right-angled triangle are 3 cm and 4 cm. This triangle is revolved around the side of length 3 cm. Find the volume of the cone thus formed. Find also the slant height of the cone (π =3.14).

Therefore by pythagorous theorem third side (slant height) will be

It is revolved about 3 cm side

Therefore, the cone formed will have dimensions as

Height, h = 3 cm

Radius, r = 4 cm

So, the volume of cone = (1/3)πr

^{2}h

= (1/3)(22/7)(4 x 4)(3)

= 22/7 x 16

= 352 / 7

= 50.28 cm

^{3}

Therefore, the correct option is c) 5 cm

### Write the factored form of the given expression.

Ans:- (7x - 4) (7x + 4) is the factorized form of the given expression.

Explanation :-

Given,

Write

Applying

We get ,

∴ (7x - 5)(7x + 5) is the factorized form of the given expression.

### Write the factored form of the given expression.

Ans:- (7x - 4) (7x + 4) is the factorized form of the given expression.

Explanation :-

Given,

Write

Applying

We get ,

∴ (7x - 5)(7x + 5) is the factorized form of the given expression.

### The volume of a cone is 7.8 cm^{3} and the area of its base is 3.9 cm^{2}, find its height?

Volume of cone = 7.8 cm

^{3}

Area of its base = 3.9 cm

^{2}

We have to find the height of cone

We know that

Volume of a cone = (1/3)πr

^{2}h

Area of base = πr

^{2}

Let us divide both these formulas to get the value of height

h = 6

Therefore, the height of the given cone is 6 cm

Therefore, the correct option is d) 6 cm

### The volume of a cone is 7.8 cm^{3} and the area of its base is 3.9 cm^{2}, find its height?

Volume of cone = 7.8 cm

^{3}

Area of its base = 3.9 cm

^{2}

We have to find the height of cone

We know that

Volume of a cone = (1/3)πr

^{2}h

Area of base = πr

^{2}

Let us divide both these formulas to get the value of height

h = 6

Therefore, the height of the given cone is 6 cm

Therefore, the correct option is d) 6 cm

### Dimensions of the cuboid are in the ratio of 5 : 4: 2 and the whole surface area is 684 cm^{2}, find the volume of the cuboid?

Length, breadth and height of a cuboid are in the ratio 5:4:2 and the total surface area is 684 cm^{2},

Let, the dimension of cuboid are

l = 5x,

b = 4x

and h = 2x

Now,

Surface area of cuboid = 2((5x 4x) + (4x2x) + (2x5x))

= 2((20x^{2})+(8x^{2})+(10x^{2}))

= 2(38 x^{2})

= 76x^{2}

Surface area of cuboid = 684

76x^{2 }= 684

x^{2 }= 9

Now,

x = 3

Dimensions of cuboid are,

l = 5x = 53 =15cm,

b = 4x = 43 = 12cm

h = 2x = 23 = 6cm

Volume of cuboid = l b h

= 15 12 6

= 1080 cm^{3}

Hence, the volume of cuboid is 1080 cm^{3} is the answer.

Therefore , the correct option is a)1080

### Dimensions of the cuboid are in the ratio of 5 : 4: 2 and the whole surface area is 684 cm^{2}, find the volume of the cuboid?

Length, breadth and height of a cuboid are in the ratio 5:4:2 and the total surface area is 684 cm^{2},

Let, the dimension of cuboid are

l = 5x,

b = 4x

and h = 2x

Now,

Surface area of cuboid = 2((5x 4x) + (4x2x) + (2x5x))

= 2((20x^{2})+(8x^{2})+(10x^{2}))

= 2(38 x^{2})

= 76x^{2}

Surface area of cuboid = 684

76x^{2 }= 684

x^{2 }= 9

Now,

x = 3

Dimensions of cuboid are,

l = 5x = 53 =15cm,

b = 4x = 43 = 12cm

h = 2x = 23 = 6cm

Volume of cuboid = l b h

= 15 12 6

= 1080 cm^{3}

Hence, the volume of cuboid is 1080 cm^{3} is the answer.

Therefore , the correct option is a)1080

### Radius and slant height of a cone are 20 cm and 29 cm respectively. Find its volume

^{2}h

By applying Pythagoras theorem on the cone, we can find the relation between volume and slant height of the cone.

We know, h

^{2}+ r

^{2}= L

^{2}

where,

- h is the height of the cone,
- r is the radius of the base, and,
- L is the slant height of the cone.

Solution:- We have given that

Radius, r = 20 cm

Slant height , L = 29 cm

Therefore, volume of cone =

= (1/3)(3.14)(20 x 20)

= (1/3)(3.14)(400) √(841 - 400)

= (1/3)(1256) √(441)

= (1/3)(1256)(21)

= 1256 x 7

= 8792 cm

^{3}

Therefore, the volume of given cone is 8792 cm

^{3}

Therefore option c) 8792 cm

^{3}is correct.

### Radius and slant height of a cone are 20 cm and 29 cm respectively. Find its volume

^{2}h

By applying Pythagoras theorem on the cone, we can find the relation between volume and slant height of the cone.

We know, h

^{2}+ r

^{2}= L

^{2}

where,

- h is the height of the cone,
- r is the radius of the base, and,
- L is the slant height of the cone.

Solution:- We have given that

Radius, r = 20 cm

Slant height , L = 29 cm

Therefore, volume of cone =

= (1/3)(3.14)(20 x 20)

= (1/3)(3.14)(400) √(841 - 400)

= (1/3)(1256) √(441)

= (1/3)(1256)(21)

= 1256 x 7

= 8792 cm

^{3}

Therefore, the volume of given cone is 8792 cm

^{3}

Therefore option c) 8792 cm

^{3}is correct.

### The area of a base of a cuboid is 48 cm^{2} and its height and length of the diagonal are 3 cm and 13 cm respectively. Calculate the length and width of the box?

^{2}and

length of diagonal =

So accordingly

(a)(b)=48 ……………..(1) and

a^{2}+b^{2}=160-------------(2).

Now using formula. (a + b)^{2 }= a^{2}+b^{2}+2.ab

(a + b)^{2 }= 160 + 2 × 48

(a + b)^{2 }= 160 + 96 = 256

Taking square root of both sides

or. a + b = 16……………(3)

and. (a - b)^{2 }= a^{2 }+ b^{2 }- 2.ab

( a- b)^{2 }= 160 – 96 = 64

Taking square root of both sides

or. a - b = 8…………………(4). ,

By adding eqn. (3) and (4) we get,

2a = 24.

a = 24/2 = 12 m.

Putting a = 12 in eqn. (3)

12 + b = 16

b = 16 – 12

b = 4 m.

Thus , length =12 m , width = 4 m.

Therefore, the correct option is c)12cm , 4cm.

### The area of a base of a cuboid is 48 cm^{2} and its height and length of the diagonal are 3 cm and 13 cm respectively. Calculate the length and width of the box?

^{2}and

length of diagonal =

So accordingly

(a)(b)=48 ……………..(1) and

a^{2}+b^{2}=160-------------(2).

Now using formula. (a + b)^{2 }= a^{2}+b^{2}+2.ab

(a + b)^{2 }= 160 + 2 × 48

(a + b)^{2 }= 160 + 96 = 256

Taking square root of both sides

or. a + b = 16……………(3)

and. (a - b)^{2 }= a^{2 }+ b^{2 }- 2.ab

( a- b)^{2 }= 160 – 96 = 64

Taking square root of both sides

or. a - b = 8…………………(4). ,

By adding eqn. (3) and (4) we get,

2a = 24.

a = 24/2 = 12 m.

Putting a = 12 in eqn. (3)

12 + b = 16

b = 16 – 12

b = 4 m.

Thus , length =12 m , width = 4 m.

Therefore, the correct option is c)12cm , 4cm.

### Internal length, height, and breadth of an open box are 95 cm, 82 cm, and 75 cm. Thickness of wood is given to be 2.5 cm, calculate the cost of painting outside the box as Rs 1 per dm^{2}

length = 95 cm,

breadth = 75 cm

height = 82 cm

Outer dimensions of the box are

length = 95 + 2 2.5 = 100 cm,

breadth = 75 + 2 2.5 = 80 cm

height = 82 + 2.5 = 84.5 --as the box is open we will add only 2.5

Surface area of the box = 2(lb+ bh +lh)

= 2((95)(75)+(75)(82)+(95)(82))

= 2(8000+6760+8450)=

= 46420 cm^{2}

Surface area of the open box = Total area - area of open side

= 46420 – (95)(75)

= 46420-8000

= 38420 cm^{2}

= 384.2 dm^{2}

Cost of painting = Rs. 384.20

Therefore option d) 384.20 is correct.

### Internal length, height, and breadth of an open box are 95 cm, 82 cm, and 75 cm. Thickness of wood is given to be 2.5 cm, calculate the cost of painting outside the box as Rs 1 per dm^{2}

length = 95 cm,

breadth = 75 cm

height = 82 cm

Outer dimensions of the box are

length = 95 + 2 2.5 = 100 cm,

breadth = 75 + 2 2.5 = 80 cm

height = 82 + 2.5 = 84.5 --as the box is open we will add only 2.5

Surface area of the box = 2(lb+ bh +lh)

= 2((95)(75)+(75)(82)+(95)(82))

= 2(8000+6760+8450)=

= 46420 cm^{2}

Surface area of the open box = Total area - area of open side

= 46420 – (95)(75)

= 46420-8000

= 38420 cm^{2}

= 384.2 dm^{2}

Cost of painting = Rs. 384.20

Therefore option d) 384.20 is correct.

### The total surface area of a cube is 846 cm^{2}. Find the height, breadth, and length if the dimensions are in the ratio of 3:4:5

^{2}

And the ratio of height breadth and length is 3:4:5

Let , height = 3x,

Breadth =4x,

Length = 5x

Total Surface Area(TSA) of cuboid = 2[ lb + bh + hl ]

2(47x

^{2}) = 846

Divide both sides by 2,

Divide both sides by 47,

Taking square root

The dimensions of cuboid are,

Height = 3x=3(3)= 9

Breadth= 4x =4(3) =12

Length = 5x = 5(3)= 15

Therefore the option a)15cm , 12cm , 9cm is correct.

### The total surface area of a cube is 846 cm^{2}. Find the height, breadth, and length if the dimensions are in the ratio of 3:4:5

^{2}

And the ratio of height breadth and length is 3:4:5

Let , height = 3x,

Breadth =4x,

Length = 5x

Total Surface Area(TSA) of cuboid = 2[ lb + bh + hl ]

2(47x

^{2}) = 846

Divide both sides by 2,

Divide both sides by 47,

Taking square root

The dimensions of cuboid are,

Height = 3x=3(3)= 9

Breadth= 4x =4(3) =12

Length = 5x = 5(3)= 15

Therefore the option a)15cm , 12cm , 9cm is correct.

### A cube of 11 cm edge is immersed completely in a rectangular vessel containing the liquid. If the dimensions of the base of the vessel are 12 cm and 15 cm, calculate the rise in the water level?

- We know that,

l =15cm

b = 12cm

- Consider the rise in the water level = h cm
- Volume of given cube = (11)
^{3}=1331 - So the volume of cube = volume of the vessel
- Substituting these values

1331 = 15 × 12 × h

- By further calculation

h = 1331/ (15 × 12)

- So we get

h = 1331/ 180 = 7.39 cm

- Therefore, the rise in the water level is 7.39 cm.
- Therefore the correct option is b)7.39.

### A cube of 11 cm edge is immersed completely in a rectangular vessel containing the liquid. If the dimensions of the base of the vessel are 12 cm and 15 cm, calculate the rise in the water level?

- We know that,

l =15cm

b = 12cm

- Consider the rise in the water level = h cm
- Volume of given cube = (11)
^{3}=1331 - So the volume of cube = volume of the vessel
- Substituting these values

1331 = 15 × 12 × h

- By further calculation

h = 1331/ (15 × 12)

- So we get

h = 1331/ 180 = 7.39 cm

- Therefore, the rise in the water level is 7.39 cm.
- Therefore the correct option is b)7.39.

### The circumference of the base of a 12 cm high wooden solid cone is 44 cm. Find the volume

^{2}h

Circumference of base = 2πr

Solution :- We have given the dimensions of a wooden solid cone

Circumference = 44 m = 2πr

44 = 2 () r

r = (44 x 7) / 44

r = 7

Radius, r = 7 m

Height, h = 12 m

We have to find the volume of the given cone

We know that

Volume of a cup = (1/3)πr

^{2}h

= ()(3.14)(7 x 7)(12)

= ()(3.14) (49 x 12)

= ()(3.14)(588)

= 1848 / 3

= 616 cm

^{3}

Therefore, the volume of wooden cone is 616 cm

^{3}

Therefore correct option is a) 616 cm

^{3}.

### The circumference of the base of a 12 cm high wooden solid cone is 44 cm. Find the volume

^{2}h

Circumference of base = 2πr

Solution :- We have given the dimensions of a wooden solid cone

Circumference = 44 m = 2πr

44 = 2 () r

r = (44 x 7) / 44

r = 7

Radius, r = 7 m

Height, h = 12 m

We have to find the volume of the given cone

We know that

Volume of a cup = (1/3)πr

^{2}h

= ()(3.14)(7 x 7)(12)

= ()(3.14) (49 x 12)

= ()(3.14)(588)

= 1848 / 3

= 616 cm

^{3}

Therefore, the volume of wooden cone is 616 cm

^{3}

Therefore correct option is a) 616 cm

^{3}.

### Given length, breadth, and height of a cuboid are 15 cm, 37.5 cm, and 48 cm. Find the cube edge whose given volume is equal to the volume of this cuboid. Also, calculate the length of the diagonal to the nearest natural number.

Length = l = 15cm

Breadth = b =37.5cm

Height = h = 48cm

Volume of a cuboid = l × b × h =

= 15 × 37.5 × 48

= 27000 cm

^{3}

As from the given condition ,

Volume of cube = volume of cuboid

a

^{3 }= 27000

a =

a = 30

Therefore the length of diagonal = a

= 30

= (1.734 )30

= 51.9

= 52

Therefore, the correct answer is d) 30cm , 52cm

### Given length, breadth, and height of a cuboid are 15 cm, 37.5 cm, and 48 cm. Find the cube edge whose given volume is equal to the volume of this cuboid. Also, calculate the length of the diagonal to the nearest natural number.

Length = l = 15cm

Breadth = b =37.5cm

Height = h = 48cm

Volume of a cuboid = l × b × h =

= 15 × 37.5 × 48

= 27000 cm

^{3}

As from the given condition ,

Volume of cube = volume of cuboid

a

^{3 }= 27000

a =

a = 30

Therefore the length of diagonal = a

= 30

= (1.734 )30

= 51.9

= 52

Therefore, the correct answer is d) 30cm , 52cm

### Vtotal = Vcone 1+Vcone 2 Find the volume of the composite solid.

Use 3.14 for pi.

For Cone 1

Radius, r = 6 cm

Height, h = 7 cm

For Cone 2

Radius, R = 10 cm

Height, H = 6 cm

We have to find the volume of the given cone

We know that

Volume of a Cone 1 = ()πr

^{2}h

= ()(3.14)(6 x 6)(7)

= ()(3.14) (36 x 7)

= ()(3.14)(252)

= 791.28 / 3

= 263.76 cm

^{3}

Volume of a Cone 1 = ()πR

^{2}H

= ()(3.14)(10 x 10)(6)

= ()(3.14) (100 x 6)

= ()(3.14)(600)

= 188

= 628 cm

^{3}

We know that V

_{total}=V

_{cone 1}+V

_{cone 2}

V

_{total }= 263.76 + 628 = 891.76 cm

^{3}

Therefore, the total volume of the composite solid is 891.76 cm

^{3}

Therefore correct option is a)891.76 cm

^{3}.

### Vtotal = Vcone 1+Vcone 2 Find the volume of the composite solid.

Use 3.14 for pi.

For Cone 1

Radius, r = 6 cm

Height, h = 7 cm

For Cone 2

Radius, R = 10 cm

Height, H = 6 cm

We have to find the volume of the given cone

We know that

Volume of a Cone 1 = ()πr

^{2}h

= ()(3.14)(6 x 6)(7)

= ()(3.14) (36 x 7)

= ()(3.14)(252)

= 791.28 / 3

= 263.76 cm

^{3}

Volume of a Cone 1 = ()πR

^{2}H

= ()(3.14)(10 x 10)(6)

= ()(3.14) (100 x 6)

= ()(3.14)(600)

= 188

= 628 cm

^{3}

We know that V

_{total}=V

_{cone 1}+V

_{cone 2}

V

_{total }= 263.76 + 628 = 891.76 cm

^{3}

Therefore, the total volume of the composite solid is 891.76 cm

^{3}

Therefore correct option is a)891.76 cm

^{3}.

### Given edges of the cubes as 3 cm, 4 cm, and 5 cm respectively. Its now melted and made into one single cube. Calculate the edge of a new cube.

x= 3cm

y=4cm

z= 5cm

And let the side if final cube is w

We will first add the volumes of the three cubes A, B and C

Vol(A) + Vol(B) + Vol(C) = x

^{3}+ y

^{3}+ z

^{3}

= 3

^{3}+ 4

^{3}+ 5

^{3}

= 27 + 64 + 125

= 216

The sum of volumes of cubes A , B and C is equal to the resultant cube we obtain by melting these three,

Volume of final cube = Vol(A) + Vol(B) + Vol(C)

(w)

^{3 }= 216

We know that 216 is the cube of 6

Therefore, w = 6

Therefore, the correct option is b) 6 .

### Given edges of the cubes as 3 cm, 4 cm, and 5 cm respectively. Its now melted and made into one single cube. Calculate the edge of a new cube.

x= 3cm

y=4cm

z= 5cm

And let the side if final cube is w

We will first add the volumes of the three cubes A, B and C

Vol(A) + Vol(B) + Vol(C) = x

^{3}+ y

^{3}+ z

^{3}

= 3

^{3}+ 4

^{3}+ 5

^{3}

= 27 + 64 + 125

= 216

The sum of volumes of cubes A , B and C is equal to the resultant cube we obtain by melting these three,

Volume of final cube = Vol(A) + Vol(B) + Vol(C)

(w)

^{3 }= 216

We know that 216 is the cube of 6

Therefore, w = 6

Therefore, the correct option is b) 6 .

### Mike has a large plastic cup that he is going to fill with water. The plastic cup is in the shape of a cone as shown. Which is closest to the volume of Mike’s cup?

^{2}h

Solution :- We have given the dimensions of a large plastic cup of Mike in shape of cone

Radius, r = 3 in

Height, h = 7 in

We have to find the volume of the given cone

We know that

Volume of a cup = ()πr

^{2}h

= ()(3.14)(3 x 3)(7)

= ()(3.14) (9 x 7)

= ()(3.14)(63)

= 197.

= 65.94 in

^{3}

= 66 in

^{3}

Therefore, the volume of Mike’s cup is 66 in

^{3}

Therefore correct option is c) 66 in

^{3}.

### Mike has a large plastic cup that he is going to fill with water. The plastic cup is in the shape of a cone as shown. Which is closest to the volume of Mike’s cup?

^{2}h

Solution :- We have given the dimensions of a large plastic cup of Mike in shape of cone

Radius, r = 3 in

Height, h = 7 in

We have to find the volume of the given cone

We know that

Volume of a cup = ()πr

^{2}h

= ()(3.14)(3 x 3)(7)

= ()(3.14) (9 x 7)

= ()(3.14)(63)

= 197.

= 65.94 in

^{3}

= 66 in

^{3}

Therefore, the volume of Mike’s cup is 66 in

^{3}

Therefore correct option is c) 66 in

^{3}.