Question 1: What is the least number of solid metallic spheres, each of diameter, that should be melted and recast to form a solid metal cone whose height is and diameter ?

Answer:

Radius of Sphere

Metal cone: Height , Radius

Let the number of cones melted

Question 2: A largest sphere is to be carved out of a right circular cylinder of radius and height . Find the volume of the sphere. (Answer correctly to the nearest integer).

Answer:

Cylinder: Radius , Height

Radius of sphere will be

Question 3: A right circular cylinder having diameter and height is full of ice-cream. The ice-cream is to be filled in identical cones of height and diameter having a hemi-spherical shape on the top. Find the number of cones required.

Answer:

Cylinder: Radius , Height

Volume of cylinder

Question 4: A solid is in the form of a cone standing on a hemisphere with both their radii being equal to and the height of the cone is equal to its radius. Find, in terms of , the volume of the solid.

Answer:

Radius

Question 5: The diameter of a sphere is . It is melted and drawn into a wire of diameter . Find the length of the wire.

Answer:

Sphere: Radius

Wire (cylinder): Radius , Length

Volume of sphere Volume of wire

Question 6: Determine the ratio of the volume of a cube to that of a sphere that will exactly fit inside the cube.

Answer:

Let radius of sphere be . The the side of the cube would be

Volume of cube

Question 7: An iron pole consisting of a cylindrical portion high and of base diameter is surmounted by a cone high. Find the mass of the pole, given that of iron 355 has of mass (approx.) (Take )

Answer:

Cylinder (Iron Pole): Height , Radius

Cone: Height

Density

Question 8: When a metal cube is completely submerged in water contained in a cylindrical vessel with diameter , the level of water rises by . Find : (i) the length of the edge of the cube, (ii) the total surface area of the cube.

Answer:

Let side of the cube

Cylinder: Radius

Question 9: In the following diagram a rectangular platform with a semi-circular end on one side is long from one end to the other end. If the length of the half circumference is , find the cost of constructing the platform, high at the rate of .

Answer:

Please refer to the diagram

Therefore length of the rectangle

Volume of rectangle

Total volume

Total cost Rs.

Question 10: The cross-section of a tunnel is a square of side surmounted by a semicircle as shown in the adjoining figure. The tunnel ii long. Calculate (i) volume, (ii) the surface area of the tunnel (excluding the floor), and (iii) it’s floor area.

Answer:

Please refer to the diagram

i) Volume Cross Section Area Length

ii) Surface area

iii) Floor area

Question 11: A cylindrical water tank of diameter and height is being fed by a Pipe of diameter through which water flows at the rate of . Calculate, in minutes, the time it takes to fill the tank.

Answer:

Water tank: Radius , Height

Pipe: Radius

Flow of water

Volume of water tank

Question 12: Water flows, at 9 km per hour, through a cylindrical pipe of cross-sectional area . If this water is collected into a rectangular cistern of dimensions : calculate the rise in level in the cistern in 1 hour 15 minutes.

Answer:

Volume of Cistern

Total volume of flow in minutes

Height of water

Question 13: The given figure shows the cross-section of a cone, a cylinder, and a hemisphere all with the same diameter 10 cm, and the other dimensions are as shown. Calculate: (a) the total surface area, (b) the total volume of the solid and (c) the density of the material if its total weight is .

Answer:

i) Total Surface Area

Since

Question 14: A solid, consisting of a right circular cone standing on a hemisphere, is placed upright in a right circular cylinder, full of water, and touches the bottom. Find the volume of water left in the cylinder, having given that the radius of the cylinder is and its height is , the radius of the hemisphere is and the height of the cone is . Give your answer to the nearest cubic centimeter.

Answer:

Cylinder: Radius , Height

Volume of cylinder

Question 15: A metal container in the form of a cylinder is Surmounted by a hemisphere of the same radius. The internal height of the cylinder is and the internal radius is Calculate: (i) the total area of the internal surface excluding the base; (ii) the internal volume of the container in

Answer:

i) Total internal surface area

ii) Internal volume

Question 16: An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is and the height of the cylindrical part is If the diameter of the base is , find the quantity of canvas required to make the tent. Allow extra for fold and for stitching. Give your answer to the nearest

Answer:

Since

Total Surface Area

Therefore canvas required

Question 17: The total surface area of a hollow cylinder, which is open from both sides, is ; area of the base ring is and height is . Find the thickness of the cylinder.

Answer:

Total surface area

Area of the base

Height

.. … … … … i)

.. … … … … ii)

Therefore from i) we get .. … … … … iii)

Solving ii) and iii) we get

Hence thickness

Question 18: A test-tube consists of a hemisphere and a cylinder of the same radius. The volume of the water required to fill the whole tube is , and of water is required to fill the tube to a level which is below the top of the tube. Find the radius of the tube and the length of its cylindrical part.

Answer:

.. … … … … i)

.. … … … … ii)

Substituting in ii)

Question 19: A solid is in the form of a right circular cone mounted on a hemisphere. The diameter of the base of the cone. which exactly coincides with hemisphere, is and its height is . The solid is placed in a cylindrical vessel of internal radius and height . How much water, in , will be required to fill the vessel completely.

Answer:

Volume of cylinder

Therefore Quantity of water required

Question 20: A cone and a cylinder have their heights in the ratio and their diameters are in the ratio . Find the ratio between their volumes.

Answer:

Let height of the cone

Therefore the height of the cylinder

Let diameter of the cone

Therefore diameter of the cylinder

Question 21: A sphere just fits in a cylindrical vessel and the height of the cylindrical vessel is the same as the height of the sphere. Show that the curved surface area of the cylinder is the same as the curved surface area of the sphere.

Answer:

Curved surface area of cylinder

Curved surface area of sphere

Therefore they are equal.