Chapter 13: Surface Areas and Volumes

(D) (32πr³)/3

Explanation:

Given, the radius of a sphere is 2r.

The volume of a sphere = (4/3) × (radius)³

Thus, the volume of the given sphere = (4/3) × π × (2r)³

= (4/3) × π × 8 × r³

= (32 π r³/3)

Hence, option D is the correct answer.

(C) 64 cm³

Explanation:

The surface area of a cube is 96 cm²

Let the length of the cube is l cm.

Thus,

6 × l² = 96

According to formula

l² = 96/6

l² = 16

l = 4

Thus the length of the cube is 4 cm.

Volume of a cube  = l³

= 4³

= 64

Thus volume is 64 cm³ which is option c.

Hence, option C is the correct answer.

(B) 2.1 cm

Explanation:

Height of cone, h = 8.4cm

Radius of base, r = 2.1cm

Thus volume of a cone = (1/3) × π × h × r²

=(1/3) × π × 8.4 × (2.1)²

Now when it is melted to form a sphere, say of radius r1 cm, the volumes of both are going to be equal.

Volume of sphere = (4/3) × π × r₁³

Thus the radius of the sphere is 2.1cm which is option b.

Hence, option B is the correct answer.

(C) same

Explanation:

Let the radius of a cylinder = r unit

Let the height of a cylinder = h unit.

According to question,

Radius is doubled = 2r

Height is halved = h/2

Then,

Curved surface area of a cylinder = 2 × π × r × h

And according to the above condition,

Curved surface area = 2 × π × 2 × r × h/2

= 2 × π × r × h

This is same as the curved surface area of the cylinder with radius r and height h.

Hence, option C is the correct answer.

(B) πr(l+(r/4))

Explanation:

Given, radius of a cone is r/2 and slant height is 2L.

Total surface area = π × radius × slant height + π × radius²

Hence, option B is the correct answer