Shape Conversion of Solids

Q.

A metallic sphere of radius 10.5 cm is melted and then recast into smaller cones, each of radius 3.5 cm and height 3 cm, How many cones are obtained?

Q.

A metallic right circular cone 20 cm high and whose vertical angle is ${60}^{\circ }$ is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter $\frac{1}{16}cm$, find the length of the wire. [4 MARKS]

Q.

Choose the correct answer in each of the following :

A solid metallic cylinder of base radius 3 cm and height 5 cm is melted to make n solid cones of height 1cm and base radius 1mm. The value of n is

(a) 450 (b) 1350 (c) 4500 (d) 13500

Q.

The diameter of a copper sphere is $18cm,$ It is melted and drawn into a long wire of uniform cross-section. If the length of the wire is $108m,$ find its diameter.

Q.

A solid lead ball of radius $6cm$ is melted and then drawn into a wire of diameter $0.2cm$. The length of wire is

$\left(a\right)272m\left(b\right)288m\left(c\right)292m\left(d\right)296m$

Q.

A right circular cone is 3.6 cm high and the radius of its base is 1.6 cm, It is melted and recast into a right circular cone having base radius 1.2 cm. Find its height.

Q.

A hemispherical bowl of internal and external diameters $6$ cm and $10$ cm is melted into a right circular cylinder of radius $14$ cm. Find the height of the cylinder

Q.

A solid metallic sphere of diameter $21cm$ is melted and recast into a number of smaller cones, of diameter $3.5cm$ and height $3cm$. The number of cones so formed is:

1. $254$
2. $405$
3. $504$
4. $540$

Q.

A hemisphere of lead of radius 9 cm is cast into a right circular cone of height 72 cm.Find the radius of the base of the cone.

Q.

A sphere of radius 4 cm is melted and recast into the shape of a cylinder of radius 8 cm. What is the height of the cylinder formed?

1. 2.22 cm

2. 3.21 cm

3. 0.95 cm

4. 1.33 cm

Q.

$\text{A sphere of diameter 12.6 cm is melted and cast into a right circular cone of height 25.2 cm. The radius of the base of the cone is}\left(a\right)6.3cm\left(b\right)2.1cm\left(c\right)6cm\left(d\right)4cm$

Q.

A sphere of diameter 15.6 cm is melted and into a right circular cone of height 31.2 cm.Find the diameter of the base of the cone.

Q.

How many spheres 12 cm in diameter can be made from a metallic cylinder of diameter 8 cm and height 90 cm?

Q. The edge of a cube is 20 cm. How many small cubes of edge 5 cm can be formed from this cube?

1. 4
2. 32
3. 64
4. 100

Q. A wooden toy is in the form of a cone surmounted on a hemisphere. The diameter of the base of the cone is 16 cm and its height is 15 cm. Find the cost of painting the toy at Rs 7 per 100 cm².

Q.

A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm³) is needed to fill this capsule?

Q. Volume of a hemisphere isn18000πcu cm . Find its diameter

Q.

A rectangular water tank of the base $11m\times 6m$ contains water up to a height of $5m$.

If the water in the tank is transferred to a cylindrical tank of radius $3.5m$, find the height of the water level in the tank.

Q. 18. A villager has a plot of the land of the shape of the quadrilateral. The gram panchayat of the village decided to take over some portion of his plot from one corner to construct a health center. He agrees with the condition that he should be given equal amount of land in return of the adjacent plot so as to form a triangular plot. Explain how this proposal be implemented

Q. If a solid sphere of radius r is melted and cast into the shape of a solid cone of height r, then the radius of the base of the cone is

(a) 2r

(b) 3r

(c) r

(d) 4r

Q.

If a sphere is inscribed in a cube, then the ratio of the volume of the sphere to the volume of the cube is
(a) $\pi :2$
(b) $\pi :3$
(c) $\pi :4$
(d) $\pi :6$

Q.

Find the total surface area of a cuboid which is $6cm$ long, $7cm$ wide and $8cm$ high.

Q.

The diameter of a sphere is 6cm. It is melted and drawn into a wire of diameter 2 mm. Find the length of the wire.

Q.

A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of Rs 498.96. If the cost of white-washing is Rs 2.00 per square meter, find the

(i) inside surface area of the dome,

(ii) volume of the air inside the dome.

Q. In Fig. 10, a right circular cone of diameter r cm and height 12 cm rests on the base of a right circular cylinder of radius r cm. Their bases are in the same plane and the cylinder is filled with water upto a height of 12 cm. If the cone is then removed, find the height to which water level will fall.

Q. If the radius and height of a cone both are increased by 10%, then the volume of the cone is increased by:

(a) 10%

(b) 21%

(c) 33.1%

(d) 100%

Q. A storage tank consists of a circular cylinder with a hemisphere adjoined on either end. If the external diameter of the cylinder by 1.4 m and its length be 8 m, find the cost of painting it on the outside at the rate of Rs. 10 per m².

Q. A hemi-spherical dome of a building needs to be painted. If the circumference of the base of the dome is 17.6 cm, find the cost of painting it, given the cost of painting is Rs 5 per 100 cm².

Q.

There is a thin coin of radius 2K mm and thickness 1 mm. Now I place M coins one on top of the other. What is the object formed and what is its volume?

1. Cylinder, 2πK²

2. Cuboid, 4πK²

3. Sphere, 2πMK²

4. Cylinder, 4πMK²

Q. Question 4
A conical tent is 10 m high and the radius of its base is 24 m. Find
(i) Slant height of the tent.
(ii) Cost of the canvas required to make the tent, if the cost of $1m^2$ canvas is Rs 70. $[Assume\pi =\frac{22}{7}]$