Chapter 13: Surface Areas and Volumes

True

Justification:

Let the radius of the sphere = r.

According to the question,

height and diameter of cylinder = diameter of sphere.

So, the radius of the cylinder = r

And, the height of the cylinder = 2r

We know that,

Volume of sphere = 2/3 volume of cylinder

Hence, the given statement “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.

False

Justification:

Let the original radius of the cone = r

Let height of the cone = h.

The volume of cone = 1/3 πr²h

Now, when radius of a height circular cone is halved and height is doubled, then

We can observe that the new volume = half of the original volume.

Hence, the given statement “if the radius of a right circular cone is halved and height is doubled, the volume will remain unchanged” is false.

Consider a right circular cone, with

Height = h

Radius = r

And slant height = l

We know, right triangle = one angle 90°

Using Pythagoras theorem in

⇒ l² = h² + r²

This justifies that height, radius and slant height of cone can always be the sides of a right triangle.

Hence, the given statement “in a right circular cone, height, radius and slant height do not always be sides of a right triangle” is true.

True

Justification:

Let radius of the cylinder = r

Height of the cylinder = h

Then, curved surface area of the cylinder, CSA = 2πrh

According to the question,

Radius is doubled and curved surface area is not changed.

New radius of the cylinder, R = 2r

New curved surface area of the cylinder, CSA’ = 2πrh …(i)

Alternate case:

When R = 2r and CSA’ = 2πrh

But curved surface area of cylinder in this case, CSA’= 2πRh = 2π(2r)h = 4πrh …(ii)

Comparing equations (i) and (ii),

We get,

Since, 2πrh ≠ 4πrh

equation (i) ≠ equation (ii)

Thus, if h = h/2 (height is halved)

Then,

CSA’ = 2π(2r)(h/2) = 2πrh

Hence, the given statement “If the radius of a cylinder is doubled and its curved surface area is not changed, the height must be halved” is true.

According to the question,

Edge of cube, l = 2r

Then, diameter of the cone = 2r

⇒ radius of the cone = 2r/2

= r

Height of the cone, h = height of the cube

= 2r

Volume of the cone is given by,

Volume of cone = 1/3 πr²h

= 1/3 πr²(2r)

= 2/3 πr³

= Volume of hemisphere of radius r

Hence, the given statement “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 true.