Question 1: From a solid right circular cylinder with height and radius of the base is , a right circular cone of the same height and the same base is removed. Find the volume of the remaining solid.

Answer:

Cylinder: Height

Cone: Height

Remaining Volume = Volume of Cylinder – Volume of Cone

Question 2: From a solid cylinder whose height is and radius is , a conical cavity of height and of base radius is hollowed out” Find the volume and total surface area of the remaining solid.

Answer:

Cylinder: Height

Cone: Height

Remaining Volume = Volume of Cylinder – Volume of Cone

Surface Area calculations:

Area of the top of the cylinder

Curved surface area of the cylinder

Curved Surface Area of Cone

Area of the base of the cone

Therefore the total surface area

Question 3: A circus tent is cylindrical to a height of and conical above it. If its diameter is and its slant height is , calculate the total area of canvas required. Also, find the total cost of the canvas at if the width if .

Answer:

Cylinder: Height

Cone: Slant Height

Curved surface area of the cylinder

Curved Surface Area of Cone

Total Surface Area of the tent

Length of the canvas needed

Cost of the total canvas

Question 4: A circus tent is cylindrical to a height of surmounted by a conical part. If total height of the tent is and the diameter of its base is ; calculate: (i) total surface area of the tent, (ii) area of canvas, required to make this tent allowing of the canvas used for folds and stitching.

Answer:

Cylinder:

Cone:

(i) Curved surface area of the cylinder

Curved Surface Area of Cone

Total Surface area =

(ii) Area of canvas

Question 5: A cylindrical boiler, high, is in diameter. It has a hemispherical lid. Find the volume of its interior, including the part covered by the lid.

Answer:

Cylinder:

Hemisphere:

Volume

Question 6: A vessel is a hollow cylinder fitted with a hemispherical bottom of the same base. The depth of the cylindrical part is and diameter of hemisphere is . Calculate the capacity and the internal surface area of vessel.

Answer:

Cylinder:

Hemisphere:

Volume

Total Internal surface area

Question 7: A wooden toy is in the shape of a cone mounted on a cylinder as shown alongside. If the height of the cone is , the total height of the toy is and the radius of the base of the cone = twice the radius of the base of the cylinder ; find the total surface area of the toy. [Take )

Answer:

Cylinder:

Cone:

Curved surface area of the cylinder

Curved Surface Area of Cone

Area of the base of the Cylinder

Area of the base of the cone

Therefore the total surface area of the toy

Question 8: A cylindrical container with diameter of base contains sufficient water to submerge a rectangular solid of iron with dimensions . Find the rise in level of the water when the solid is submerged.

Answer:

Cylinder:

Volume of iron solid

Rise in water level

Question 9: Spherical marbles of diameter are dropped into beaker containing some water and are fully submerged. The diameter of the beaker is . Find how many marbles have been dropped in it if the water rises by .

Answer:

Sphere: Radius

Beaker: Radius

No of marbles

Question 10: The cross-section of a railway tunnel is a rectangle broad and high surmounted by a semi-circle as shown in: the figure. The tunnel is long. Find the cost of plastering the internal surface of the tunnel (excluding the floor) at the rate of .

Answer:

Rectangle: Breadth

Semi-circle: Radius

Length of the tunnel

Surface Area of the tunnel

Cost of plastering

Question 11: The horizontal cross-section of a water tank is in the shape of a rectangle with semi-circle at one end, as shown in the following figure. The water is deep in the tank. Calculate the volume of the water in the tank in gallons.

Answer:

Rectangle: Width

Semi-circle: Radius

Depth of the water

Cross section of the tank

Therefore, water in the tank

Hence the volume of water in gallons

Question 12: The given figure shows the cross-section of a water channel consisting of a rectangle and a semi-circle. Assuming that the channel is always full, find the volume of water discharged through it in one minute if water is flowing at the rate of . Give your answer in cubic meters correct to one place of decimal.

Answer:

Rectangle: Length

Semi-circle: Radius

Cross Section area

Amount of water flowing

Question 13: An open cylindrical vessel of internal diameter and height stands on a horizontal table. Inside this is placed a solid metallic right circular cone, the diameter of whose base is and height . Find the volume of water required to fill the vessel. If this cone is replaced by another cone, whose height is and the radius of whose base is , find the drop in the water level.

Answer:

Cylinder:

Cone:

Volume of water required

If the cone was replaced by another cone with , then let the drop in water level . Therefore

Question 14: A cylindrical can, whose base is horizontal and of radius , contains sufficient water so that when a sphere is placed in the can, the water just covers the sphere. Given that the sphere just fits into the can, calculate: (i) the total surface area of the can in contact with water when the sphere is in it; (ii) the depth of water in the can before the sphere was Put into the can.

Answer:

Cylinder:

Sphere:

Total Surface Area = Curved Surface area of Cylinder + Area of the base

Let the depth of the water

Therefore:

Question 15: A hollow cylinder has solid hemisphere inward at one end and on the other end it is closed with a flat circular plate. The height of water is when flat circular surface is downward. Find the level of water, when it is inverted upside down, common diameter is and height of the cylinder is .

Answer:

Cylinder:

Sphere:

Volume of water

Let the height of water when cylinder is upside down