*Notes: Important formulas to be kept in mind:*

*Parameters of a Cone: Radius of the base (), Height of the cone () and Slant Height of a Cone ()*

*Volume of a Cone *

*Curved Surface area of a Cone *

*Total Surface area of a Cone *

Question 1: Find the volume of a cone whose slant height is and radius of base is .

Answer:

Volume of a Cone

Therefore

Therefore Volume

Question 2: The curved surface area of a cone is . If the radius of its base is , find its height.

Answer:

Curved surface area of the cone

Therefore

Question 3: The circumference of the base of a high conical tent is . Find the volume of the air contained in it.

Answer:

Circumference of the base

Therefore

Volume of a Cone

Question 4: The radius and the height of a right circular cone are in the ratio and its volume is . Find the radius and slant height of the cone. (Take )

Answer:

Given: Radius and the height of a right circular cone are in the ratio

Let and

Volume

Therefore:

Therefore Radius and Height

Question 5: Two right circular cones and, are made. having three times the radius of and having half the volume of . Calculate the ratio between the heights of and .

Answer:

Let the radius of cone Y is . Therefore the radius of cone X .

Let the height of cone Y and the height of cone X

Given:

Question 6: The diameters of two cones are equal. If their slant heights are in the ratio , find the ratio of their curved surface areas.

Answer:

Let the radius of Cone 1 and Cone 2

Let the slant height of the cones be and respectively.

Curved Surface Area of a cone

Therefore

Question 7: There are two cones. The curved surface area of one is twice that of the other. The slant height of the latter is twice that of the former. Find the ratio of their radii.

Answer:

Let the curved surface area of the cones be and respectively.

Similarly, let the slant height of the two cones be and respectively.

Similarly, let the radius of the two cones be and respectively.

Given:

Therefore

Question 8: A heap of wheat is in the form of a cone of diameter and height . Find its volume. How much cloth is required to just cover the heap?

Answer:

Diameter

Radius

Height

Volume of a Cone

Curved Surface area of a Cone

Question 9: Find what length of canvas, in width, is required to make a conical tent in diameter and in height. Given that of the canvas is used in folds and stitching. Also, find the cost of the canvas at the rate of per meter.

Answer:

Diameter

Height

Curved Surface Area

Therefore the length of the canvas needed

However, of the canvas is consumed in folds and stitches. Hence the length needed

The total cost of the canvas

Question 10: A solid cone of height and base radius is melted and recast into identical cones, each of height and diameter . Find the number of cones formed.

Answer:

Large Cone:

Small Cones:

Therefore the number of cones made

Question 11: The total surface area of a right circular cone of slant height is . calculate: (i) its radius in . (ii) its volume in . [Take ]

Answer:

Slant length

Let the Radius

Total Surface Area

Therefore

is not possible.

Hence Height

Volume

Question 12: The area of the base of a conical solid is and its volume is . Find the curved surface area of the solid.

Answer:

Area of the base

Volume of the cone

Therefore

Therefore Slant Length

Curved Surface Area

Question 13: A vessel in the form of an inverted cone, is filled with water to the brim. Its height is and diameter of the base is . Six equal solid cones are dropped in it, so that they are fully submerged. As a result. one-fourth of water in the original cone overflows. what is the volume of each of the solid cones submerged?

Answer:

Height

Diameter

Therefore Radius

Let the volume of each of the solid cone be

Therefore:

Question 14: The volume of a conical tent is and the area of the base floor is . 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 . [2008]

Answer:

Volume

Area of the base

(i)

(ii)

(iii) Curved Surface Area

Length of canvas needed