Surface Areas and Volumes: 9 Math

NCERT Exemplar Exercise 13.2: 9th math

Write True or False and justify your answer in each of the following:

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (1) The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.

Answer True

Explanation

Given, height and diameter of a cylinder = height and diameter of a sphere

Thus, find the volume of a sphere is equal to two-third of the volume of the cylinder or not.

Let, diameter of the given sphere = 2r

Thus, radius of the sphere = r

Therefore, height of the sphere will also be equal to 2r

[Since, height and diameter of the sphere is equal. And height is called the diameter of a sphere]

And, as given in the question, height and diameter of the cylinder = height and diameter of the sphere

Thus, height (h) of the cylinder = 2r

And, diameter of the cylinder = 2r

Therefore, radius of the cylinder = r

Now, we know that Volume of a Sphere

= 4/3 ℼ r³ - - - - (i)

And, Volume of a cylinder

= ℼ r²h

Therefore, volume of the given cylinder

= ℼ r² 2r

= ℼ 2 r³ - - - - (ii)

Now, as per question,

= Volume of the given sphere = 2/3 of volume of the given cylinder

⇒ 4/3 ℼ r³ = 2/3 ℼ 2r³h

[From equation (i) and (ii)]

⇒ 4/3 ℼ r³ = 4/3 ℼ r³

And, hence the volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere. is true.

Thus, answer = true.

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (2) If the radius of a right circular cone is halved and height is doubled, the volume will remain unchanged.

Answer False

Explanation

Let, radius of the given right circular cone = r

And, height of the given right circular cone = h

And, if the radius of the given right circular cone = r/2

And, height of the given right circular cone = 2h

Thus, volume of the given right circular cone will be remain unchanged or not?

Here, we know that, Volume of a right circular cone

= 1/3 ℼ r² h - - - - (i)

Thus, after halved of the radius and doubled of the height, the volume of the given right circular cone

= 1/3 ℼ (r/2)² × 2h

= 1/3 ℼ `r^2/4` × 2h

= 1/3 ℼ `r^2/2` h - - - - - (ii)

Now, since, equation (i) ≠ equation (ii),

Thus, given quote in the question is false.

Thus, Answer = False

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (3) In a right circular cone, height, radius and slant height do not always be sides of a right triangle.

Answer False

Solution

We know that, height, radius and slant height of a right circular cone always make a right angle triangle.

Thus, the quote in the question, "In a right circular cone, height, radius and slant height do not always be sides of a right triangle" is false.

Hence, Answer = False

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (4) If the radius of a cylinder is doubled and its curved surface area is not change the height must be halved.

Answer Right

Explanation

Case-1

Let, radius of a cylinder = r

And, height of the cylinder = h

Case-2

And, when radius of the cylinder is doubled, this means radius = 2r

And, height = halved = h/2

Then, curved surface area is not changed, True or False?

Now, we know that Curved Surface Area of a Cylinder

= 2 ℼ r h

Therefore, Curved Surface Area of the given cylinder in Case-1

= 2 ℼ × r × h - - - - (i)

Now, Curved Surface Area of the given cylinder in Case-2

= 2 ℼ × 2r × h/2

= 2 ℼ r × h

= 2 ℼ r × h - - - - - (ii)

Now, clearly, equation (i) = equation (ii)

Thus, the quote given in the question, "If the radius of a cylinder is doubled and its curved surface area is not change the height must be halved." is Right.

However, if the radius is doubled and height be one-fourth of a cylinder, then its curved surface area will be halved.

Proof

Let, radius = 2r and height = h/4

Therefore, Curved Surface Area of the cylinder

= ℼ × 2r × h/4

= ℼ  r × h/2

= ℼ r × h/2 - - - - - (iii)

Now, since equation (i) ×1/2 = equation (iii)

Thus, if the radius is doubled and height be one-fourth of a cylinder, then its curved surface area will also be halved.

Thus, Answer = Right

Solution of NCERT Exemplar Exercise 13.2 Surface Areas And Volumes Class 9 Math   Question (5) The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere of radius r.

Answer Right

Explanation

Given, edge of a cube = 2r

Therefore, diameter of largest right circular cone which can be fit into the given cube having edge equal to 2r = 2r

This means the radius of the given right circular cone = r

And, height of the right circular cone = 2r

Now, we know that, Volume of a Right Circular Cone

= 1/3 ℼ r² h

Thus, volume of the given cone

= 1/3 ℼ r² 2r

= 2/3 ℼ r³ - - - - - (i)

Now, radius of the hemisphere = r

And we know that, Volume of a hemisphere

= 2/3 ℼ r³ - - - - (ii)

Here, it is clear that, equation (i) = equation (ii)

Thus, quote given in the question the volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere of radius r is Right.

Thus, Answer = Right

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