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Volume of First cuboids = 5 × 6 × 7 = 210 vm³

Volume of second cuboids = 4 × 7 × 8 = 224 cm³

Volume of third cuboids = 2 × 3 × 13 = 78 cm³

Volume of cube = 210 + 224 + 78 = 512

Let side of cube = a

⇒ a³ = 512

a = 8 cm

False

The dimensions of the given cuboid are 2 x 1 x 1. It is sliced into two equal parts, which are cubes.

Then, the dimensions of the cube, so formed are 1 x 1 x 1.

.^.. The surface area of the cube so formed = 6 (Side)² = 6 x (1)² = 6 sq units

Hence, the surface area of the sliced cube is 6 sq units.

24

Explanation: 

Given that, the cube side is 4 cm, then the volume of cube is 4³ = 64 cm³

When it is sliced into 1 cubic cm, we will get 64 small cubes

In each side of the larger cube, the smaller cubes in the edges should have more than one face painted. Therefore, the cube which are located at the corner of the larger cube, have three faces painted.

Hence, in each edge two small cubes are left, in which two faces painted.

It is known that the total numbers of edges in a cubes = 12.

Thus, the number of small cubes with two faces painted = 12 × 2 = 24 small cubes.

True

Given, a cube of side 3 cm is painted on all its faces. Now, it is sliced into 1 cu cm cubes. Then, there will be 8 corner cubes that have 3 sides painted, 6 centre cubes with only one side painted and only 1 cube in the middle that has no side painted.

The correct answer is option (c) 54

Explanation:

Given: The cube side = 5 cm

The side of cube 5 cm is cut into 5 equal parts, in which each of 1 cm

Therefore, the total number of cubes of side 1 cm = 25 + 25 + 25 + 25 + 25 = 125

In one face of cube, there are total of 9 small cubes painted.

We know that, there are 6 faces in cube.

Thus, total of 9 x 6 faces will have one face painted.

(i.e.) 54

Length (l) = m 2m and Breadth (b) = 1m 50cm 

= 91 + 50/100)m = 1.5m

Areas = 1 × 6 = 2 × 1.5 = 3m²

Length (l) of the top of the table = 2 m 25 cm = 2 + 0.25

= 2.25 m

Breadth (b) of the top of the table = 1 m 50 cm 

= 1 + 0.50 = 1.50 m

Perimeter of table – top = 2 (l + b)

= 2 × (2.25 + 1.50)

= 2 × 3.75 = 7.5 m

The perimeter of the table-top is 7.5 m

(a) perimeter of square = 4 × 25 = 100 cm.

(b) perimeter of rectangle = 2 × (10 + 40) = 2 × 50 = 100 cm

(c) perimeter of rectangle = 2 × (20 + 30) = 100 cm.

(d) perimeter of ∆le = 30 + 30 + 40 = 100 cm

Length (l) of rectangular box = 40 cm

Breadth (b) of rectangular box = 10 cm

Length o f tape required = perimeter of rectangular box

= 2 ( l + b)

= 2(40 + 10) = 100 cm

Perimeter of a polygon is equal to the sum of the lengths of all sides of that polygon.
(a) Perimeter = (4 + 2 +1 + 5) cm = 12 cm

(b) Perimeter = (23 + 35 + 40 + 35) cm = 133 cm

(c) Perimeter = (15 + 15 + 15 + 15) cm = 60 cm

(d) Perimeter = (4 + 4 + 4 + 4 +4) cm = 20 cm

(e) Perimeter = (1 + 4 + 0.5 + 2.5 + 2.5 + 0.5+ 4) cm = 15 cm

(f) Perimeter = (1 + 3 + 2 + 3 + 4 + 1 + 3 + 2 + 3 + 4 + 1 + 3 + 2 + 3 + 4 + 1 + 3 + 2 + 3 + 4) = 52 cm

(1) 4πr² sq. units

Number of cards = 84

Area of the chart paper = 240 cm² 

Correct option is: A) 1 : 8

Given that \(\frac {S_1}{S_2} = \frac 14\) 

\(\Rightarrow\) \(\frac {4\pi r_1^2}{4\pi r_2^2} = \frac 14\)

\(\Rightarrow\) \((\frac {r_1}{r_2})^2 = \frac 14 = (\frac 12)^2\) 

\(\Rightarrow\) \(\frac {r_1}{r_2}\) = \(\frac 12\) 

Now, \(\frac {V_1}{V_2} \) = \(\frac {\frac 43 \pi r^3_1}{\frac 43 \pi r^3_2} = (\frac {r_1}{r_2})^3 = (\frac 12)^3 = \frac 18\)

Hence, the ratio of their volumes is 1 : 8

Correct option is: B) 1 : 8

Correct option is: B) 38808 cm³

The radius of the sphere is r = 21 cm.

\(\therefore\) volume of the sphere = \(\frac 43 \pi r^3\)

= \(\frac 43 \times \frac {22}7 \times 21 \times 21 \times 21 = 88 \times 441\)

= 38808 \(cm^3\)

Correct option is: B) 38808 cm³

(i) side of a cube (a) = 5 cm 

Volume of a cube V = a³ 

= 5 x 5 x 5 = 125 cm³ 

(ii) a = 3.5 m

V = a³ = 3.5 x 3.5 x 3.5 m³ = 42.875 m³ 

(iii) a = 21 

V = a³ = 21 x 21 x 21 cm³ 

= 9261 cm³

Equal side is except (i) is-

(B) (ii)

(A) – (iv), (B) – (i), (C) – (ii), (D)– (iii)

Correct option is: A) R² : r²

\(\frac {V_1}{V_2} \) = \(\frac {\pi \, r_1^2h_1}{\pi \, r_2^2 h_2} = \frac {\pi R^2h}{\pi r^2h} = \frac {R^2}{r^2} = R^2 : r^2\)

Hence, the ratio of their volumes is \(R^2 : r^2\).

Correct option is: A) R² : r²

Correct option is: B) \(\frac{r}{3}\)

Volume of sphere of radius r = \(\frac 43 \pi r^3\) 

= \(\frac r3 \times 4 \pi r^2\) 

= \(\frac r3 \times \) surface area of sphere

Hence, the volume of sphere of radius 'r' is obtained by multiplying its surface area by  \(\frac r3 \)

Correct option is: B) \(\frac{r}{3}\)

Correct option is: D) 4 : 3

In cylinder, h = r

\(\frac {V_1}{V_2} \) = \(\frac {volume\, of\, sphere}{volume \,of \,cylinder } = \frac {\frac 43 \pi r^3}{\pi r^2h}\) 

= \(\frac {4r}{3h} = \frac {4r}{3r} = \frac 43\) (\(\because\) h = r)

Hence, the ratio of their volumes is 4 : 3

Correct option is: D) 4 : 3

Correct option is: A) 2r

Let radius of the base of the cone be R.

\(\because\) Volume of cone = volume of sphere

= \(\frac 13 \pi R^2 h\) = \(\frac 43 \pi r^3 h\)

= \(R^2 = \frac {4r^3}{h}\) 

= \(R^2 = \frac {4r^3}{r}\)( \(\because\) h = r)

= \(4r^2\) 

= \((2r)^2\) 

\(\therefore\) R = 2r

Hence, the radius of the base of the cone is 2r.

Correct option is: A) 2r

Correct option is: C) volume

To find out the quantity of water inside a bottle we measure the volume of bottle.

Correct option is: C) volume

Correct option is: A) 2r

Let radius of the base of the cone be R.

\(\because\) Volume of cone = volume of sphere

= \(\frac 13 \pi R^2 h\) = \(\frac 43 \pi r^3 h \)

= \(R^2 = \frac {4r^3}{h}\) 

= \(R^2 = \frac {4r^3}{r}\)( \(\because\) h = r)

= \(4r^2\) 

= \((2r)^2\) 

\(\therefore\) R = 2r

Hence, the radius of the base of the cone is 2r.

Correct option is: A) 2r

Correct option is: C) 5

Let the side of the cube be a.

\(\therefore\) Volume of the cube is \(a^3\).

But given that volume of the cube is 125 \(m^3\)

\(\therefore\) \(a^3\) = 125 \(m^3\) = \(5^3 \, m^3\)

= a = 5 m

Hence, the side of the cube is 5 m.

Correct option is: C) 5

Correct option is: D) 3

Let side of the cube be a.

\(\therefore\) T..S.A of the cube is 6 \(a^2\) 

Given that T.S.A of the cube is 54 \(cm^2\) .

\(\therefore\)   6 \(a^2\)  = 54

= \(a^2\) = \(\frac {54}6 = 9 = 3^2\) 

\(\therefore\) a = 3 cm

Hence, the side of the cube is 3 cm.

Correct option is: D) 3

Correct option is: D) \(\frac{3}{2}\) \(cm^2\)

Given that side of the cube is a = 0.5 cm

\(\therefore\) TSA of the cube = 6 \(a^2 = 6 \times (0.5)^2 = 6 \times 0.25 = 1.5 cm^2 = \frac 32 cm^2\)

Correct option is: D) \(\frac{3}{2}\) cm²

Correct option is: B) 54 cm²

The volume of a cube is \(a^3 = 27 cm^2\)

= \(a^3 = 27 = 3^3\) 

\(\therefore\) a = 3 cm

Hence, the side of the cube is 3 cm.

\(\therefore\) Surface are of cube = 6 \(a^2\)

= \(6 \times 3^2 = 6 \times 9 = 54 \, cm^2\) 

Hence, surface area of cube is 54 \(cm^2\)

Correct option is: B) 54 cm²

Correct option is: D) 7

Given that radius of cylindrical bottle is r = 2 cm.

Volume of the cylindrical bottle is V = 88 \(cm^3\).

\(\therefore\) \(\pi r^2h\) = 88

\(\Rightarrow\) h = \(\frac {88}{\pi r^2} = \frac {88\times 7}{22 \times 2 \times 2} = 7 \, cm\)

Hence, height of cylindrical bottle is h = 7 cm.

Correct option is: D) 7

The correct answer is option (a) 30 cm³

Explanation:

It is given that, the coterminous faces of a cuboid is given as:

l × b = 6

l × h = 15

b × h = 10

The formula for volume of a cuboid is l × b × h

l² × b² × h² = 6 × 15 × 10

√(lbh) = √(900) = 30