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