Surface Areas and Volumes: 9 Math

NCERT Exemplar Exercise 13.2 Q6-10: 9th math

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (6) A cylinder and a right circular cone are having the same base and same height. The volume of the cylinder is three times the volume of the cone.

Answer True

Explanation

Given, the base and height of a right circular cone and a cylinder are equal.

Thus, whether the volume of the cylinder is three times the volume of the cone or not?

In the case of right circular cone

Now, let the diameter of the right circular cone = 2r

Therefore, radius = 2r/2 = r

And, thus, height of the right circular cone = h

Now, we know that Volume of a right circular cone

= 1/3 ℼ r² h

Thus, volume of the given right circular cone

= 1/3 ℼ r² × h - - - - (i)

In the case of cylinder

Now, let the diameter of the cylinder = 2r

Therefore, radius = 2r/2 = r

And, thus, height of the cylinder = h

Now, we know that, Volume of a cylinder

= ℼ r² h

Thus, volume of the given cylinder

= ℼ r² × h - - -  - - (ii)

Now, according to question, volume of the cylinder = 3 × volume of the right circular cone

⇒  ℼ r² h = 3 × `1/3` ℼ r² h

⇒  ℼ r² h  =  ℼ r² h

Thus, quote given in the question "A cylinder and a right circular cone are having the same base and same height. The volume of the cylinder is three times the volume of the cone" is true.

Hence, Answer = True

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (7) A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1:2:3.

Answer = True

Explanation

Given, the base and height of a cone, a hemisphere and a cylinder are equal.

Thus, to prove the ratio of their volumes = 1 : 2 : 3

As given, in the question, the base and height of the cone, hemisphere and cylinder are equal.

And in the case of hemisphere, the height is equal to the radius of the hemisphere

Thus, if the radius of a hemisphere = r

Then, its height will also be equal to r.

In the case of given hemisphere

Let the diameter of the base of given hemisphere = 2r

[Because according to question, the base and height are same]

Therefore, radius of the hemisphere = r

And, since, height and radius of the hemisphere are equal

Thus, height of the hemisphere = r

Now, we know that, Volume of a hemisphere

= 2/3 ℼ r³

Thus, volume of the given hemisphere

= 2/3 ℼ r³ - - - - (i)

In the case of given right circular cone

And, let the diameter of the base of given cone = 2r

[Because according to question, the base and height are same]

Therefore, radius of the cone = r

And, since height of the cone is equal to the height of the hemisphere, thus, height of the cone = r

Now, we know that, Volume of a cone

= 1/3 ℼ r ² h

Thus, volume of the given cone

= 1/3 ℼ r² × r

= 1/3 ℼ × r³ - - - - - (ii)

In the case of given cylinder

Let the diameter of the base of given cylinder = 2r

[Because according to question, the base and height are same]

Therefore, radius of the given cylinder = r

And, since height of the cylinder and hemisphere are equal

Then, height of the cylinder = r

Now, we know that, Volume of a cylinder

= ℼ r ² h

Thus, volume of the given cylinder

= ℼ r² × r

= ℼ × r³ - - - - (iii)

Now, ratio of volumes of cone, hemisphere and cylinder

[From equation (i), (ii) and (iii)]

[After multiplying with 3]

= 1 : 2 : 3

Thus, ratios of the volumes of the given cone, hemisphere and cylinder = 1 : 2 : 3

Thus, the quote given in the question "A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1:2:3" is true.

Thus, the Answer = True

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (8) If the length of the diagonal of a cube is `6sqrt3` cm, then the length of the edge of the cube is 3 cm.

Answer False

Explanation

Given, Diagonal of the cube `=6sqrt3`

Thus, whether side of the cube = 3 cm or not?

We know that, diagonal of a cube = a √ 3

Where, a is the edge of the cube.

Thus, diagonal of the given cube,

6 √ 3 cm = a √ 3

`=>a = (6sqrt3)/(sqrt3)` cm

⇒ a = 6 cm

Thus, edge of the given cube ≠ 3 cm

Hence, Answer = False

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (9) If a sphere is inscribed in a cube , then the ratio of the volume the cube to the volume of the sphere will be 6 : ℼ.

Answer Right

Explanation

Let, edge of the given cube = a

Thus, diameter of the given sphere = a

Therefore, radius of the sphere = a/2

[Because as per question, sphere is inscribed in the cube]

Thus, is volume of the cube : volume of sphere is equal to 6 : ℼ or not?

We know that, volume of a cube = edge³

Thus, volume of the given cube = a³ - - - - (i)

And, we know that, volume of a sphere

= 4/3 ℼ r³

Thus, volume of the given sphere

Now, ratio of the volume of the cube to the volume of the sphere

`=a^3\ : 1/6\ a^3\ pi`

[From equation (i) and equation (ii)]

After multiplying both of the term of the given ratio by 6/a³

`=a^3xx6/a^3:1/6\a^3\ pi\ xx6/a^3`

= 6 : ℼ

Thus, ratio of the volume of the cube to the volume of the sphere = 6 : ℼ

Hence, Answer = Right

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (10) If the radius of a cylinder is doubled and height is halved, the volume will be doubled.

Answer True

Explanation

Given, radius of a cylinder is doubled and height is halved.

Then, whether the volume will be doubled or not?

Let, radius of the cylinder = r

And, height of the cylinder = h

Now, we know that, Volume of a cylinder

= ℼ r² h - - - - - (i)

And, after doubling of radius and halved of the height of the given cylinder,

The radius of the cylinder = 2r

And, height = h/2

Thus, volume of the given cylinder

= ℼ (2r)² × h/2

= ℼ 4 r² × h/2

= ℼ 2 r² h

= 2 ℼ r² h - - - - (ii)

Now, it is clear from equation (i) and (ii)

That, 2 × equation (i) = equation (ii)

That is volume of the cylinder in equation (i) × 2 = volume of the cylinder in equation (ii)

Thus, quote given in the question "If the radius of a cylinder is doubled and height is halved, the volume will be doubled" is true.

Hence Answer = True

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