Volume of a Right Circular Cone

Q.

Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 9 cm.

Q.

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume is $\frac{1}{27}$ of the volume of the given cone, at what height above the base is the section made?

1. 10 cm
2. 20 cm
3. 40 cm
4. 15 cm

Q.

If the slant height of a right circular cone is $3cm$ then the maximum volume of the cone is:

1. $2\sqrt{3}\mathrm{\pi }{\mathrm{cm}}^3$

2. $4\sqrt{3}\mathrm{\pi }{\mathrm{cm}}^3$

3. $\left(2+\sqrt{3}\right)\mathrm{\pi }{\mathrm{cm}}^3$

4. $\left(2-\sqrt{3}\right)\mathrm{\pi }{\mathrm{cm}}^3$

Q.

A right triangle whose sides are 15 cm and 20 cm (other than hypotenuse), is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. (Choose value of $\pi$ as found appropriate)

Q.

A solid cone of base radius 10 cm is cut into two parts through the midpoint of its height, by a plane parallel to its base. Find the ratio of the volumes of the two parts of the cone.

Q.

Find the maximum volume of a cone that can be carved out of a solid hemisphere of radius r cm.

Q.

A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.

Q. Question 5
A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

Q.

A right circular cone is divided into three parts by trisecting its height by two planes drawn parallel to the base. Show that the volumes of the three portions starting from the top are in the ratio 1 : 7 : 19.

Q.

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio fo the volume of the smaller cone to the whole cone is

(a) 1 : 2 (b) 1 : 4 (c) 1 : 6 (d) 1 : 8

Q.

Question 3
A cone is 8.4cm high and the radius of its base is 2.1cm. It is melted and recast into a sphere. The radius of the sphere is:

A) 4.2 cm
B) 2.1 cm
C) 2.4 cm
D) 1.6 cm

Q. The surface area of a solid metallic sphere is $616c{m}^2$. It is melted and recast into a smaller sphere of diameter 3.5 cm. How many can such spheres be obtained? [4 MARKS]

Q. 19. A cone of height 24 cm has a curved surface area of 550 cm2 . Find its volume.

Q. Question 3
How many cubic centimetres of iron is required to construct an open box whose external dimensions are 36 cm, 25 cm and 16.5 cm provided the thickness of the iron is 1.5 cm. If one cubic cm of iron weighs 7.5 g, find the weight of the box.

Q.

$504$ cones, each of diameter $3.5\mathrm{cm}$ and height $3\mathrm{cm}$, are melted and recast into a metallic sphere. Find the diameter of the sphere.

Q. The radius and height of a cone are in the ratio 1:2. If its volume is 144π cm${}^3$, its slant height = .

1. 5$\sqrt{5}$ cm
2. 6$\sqrt{5}$ cm
3. 4$\sqrt{5}$ cm

Q.

A milk container is made of metal sheet in the shape of frustum of a cone whose volume is $10459\frac{3}{7}c{m}^3.$ The radii of its lower and upper circular ends are 8 cm and 20 cm respectively. Find the cost of metal sheet used in making the container at the rate of Rs 1.40 per $c{m}^2$.

Q.

The radii of the circular ends of a bucket of height 15 cm are 14 cm and r cm (r < 14). If the volume of bucket is 5390 $c{m}^3$, find the value of r.

Q.

A right cylinderical vessel is full of water. How many right cones having the same radius and height as those of the right cylinder will be needed to store that water?

Q.

The radii of the ends of a frustum of a cone 45 cm high are 28 cm and 7 cm. Find its volume, the curved surface area and the total suface area (Take $\pi$= $\frac{22}{7}$))

Q.

Find the volume of a right circular cone that has a height of $5.7\mathrm{m}$ and a base with a circumference of $2.3\mathrm{m}$. Round your answer to the nearest tenth of a cubic meter.

Q.

The radii of the bases of two solid right circular cones of same height are $r_1$ and $r_2$ respectively. The cones are melted and recast into a solid sphere of radius R. Find the height of each cone in terms of $r_1,r_2$ and R.

Q.

The volume of a conical tent is 1232 $m^3$ and the area of the base floor is 154 $m^2$. Calculate the :

(i) radius of the floor,

(ii) height of the tent,

(iii) length of the canvas required to cover this conical tent if its width is 2 m.

Q. (a) A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate the number of cones recast.

(b) A hollow sphere of internal and external radii 6 cm and 8 cm respectively is melted and recast into small cones of base radius 2 cm and height 8 cm. Find the number of cones. [6 MARKS]

Q.

A solid cone of height 8 cm and base radius 6 cm is melted and recast into identical cones, each of height 2 cm and diameter 1 cm.Find the number of cones formed.

Q.

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

$\begin{array}{cc}Assertion\left(A\right)& Reason\left(R\right)\\ Thecurvedsurfaceareaofa& Volumeofacone=\pi r^{2}h.\\ coneofbaseradius3cmand& \\ height4cmis\left(15\pi \right)c{m}^2.& \end{array}$

The correct answer is (a) / (b) / (c) / (d).

Q.

A girl fills a cylindrical bucket 32 cm in height and 18 cm in radius with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24 cm, find :
(i) its radius,
(ii) its slant height. [3 MARKS]

Q. Find the volume of a sphere of radius 5.6 cm.

1. $799c{m}^3$
2. $765.45c{m}^3$
3. $700c{m}^3$
4. $735.91c{m}^3$

Q.

The volume of a cone of radius 7 $cm$ and slant height 14 $cm$ is ______.
$(use\pi =\frac{22}{7})$

1. $522.381c{m}^3$
2. $618.381c{m}^3$
3. $722.381c{m}^3$
4. $622.381c{m}^3$

Q.

The height of a cone is 20 cm. A small cone is cut off from the top by a plane parallel to the base. If its volume be 1/125 of the volume of the orginal cone, determine at what height above the base the section in made.