Geometry

Solids

Medium

If a cone and sphere have equal radii and volumes, then determine the ratio of the diameter of the sphere to the height of the cone?

3:1

1:2

6:1

1:3

✅ Correct Option: 2

A cone and sphere have equal radii $r$ and equal volumes.

Volume of sphere: $V_{sphere} = \frac{4}{3}\pi r^3$

Volume of cone: $V_{cone} = \frac{1}{3}\pi r^2 h$

where $h$ is the height of the cone.

Since the volumes are equal:

$\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h$

$4\pi r^3 = \pi r^2 h$

$4r^3 = r^2 h$

$4r = h$

The height of the cone is $4r$ .

The diameter of the sphere is $2r$ .

The height of the cone is $4r$ .

Ratio of diameter to height:

$\frac{2r}{4r} = \frac{1}{2}$

Therefore, the ratio of the diameter of the sphere to the height of the cone is $1:2$ .

2023: 24 May Shift 3

A cuboidal slab of copper having dimensions 22 cm $\times$ 10 cm $\times$ 5 cm is melted and recasted in the form of a wire of 1 mm diameter. The wire is rubber coated at the rate of Rs. 1.50 per meter. Find the cost of rubber coating (in Rs.) Use $\pi = \frac{22}{7}$ .

2025: 2 June Shift 2

A solid, metallic right circular cone of radius 7 cm and height 3 cm is melted into three cubes. If the sides of two cubes are 3 cm and 5 cm, then the side of the third cube will be how much?

2026: 18 May Shift 2

Consider the surface area of the following solids.

A. A cube of edge 9 cm.

B. A cuboid of edges 18 cm, 14 cm and 7 cm.

C. A cylinder with radius of 14 cm and height 80 cm.

D. A sphere with radius of 14 cm.

Arrange the above in decreasing order.

2026: 21 May Shift 1

Consider the surface areas of the following solids:

(A) A cuboid of 10 cm long, 7 cm broad, and 5 cm high.

(B) A cylinder of radius 3.5 cm and height 60 cm.

(D) A sphere of radius 12 cm.

Arrange the above in decreasing order.

2026: 30 May Shift 2

Consider the surface area of the following solids.

(A) A cylinder with a diameter of base 14 cm and height 80 cm.

(B) A cone with a radius of 7 cm and slant height 10 cm.

(D) A hemisphere with a radius of 21 cm.

Arrange the above in decreasing order.

Choose the correct answer from the options given below:

2026: 21 May Shift 2

Two cubes have their volumes in the ratio 1:27. The ratio of their surface areas is?

If a cone and sphere have equal radii and volumes, then determine the ratio of the diameter of the sphere to the height of the cone?

Solution

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