Advertisements
Advertisements
Question
A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the cone and of the remaining solid left out after the cone carved out.
Solution
The base of the largest right circular cone will be the circle inscribed in a face of the cube and its height will be equal to an edge of the cube.
Given, Edge of the cube = 14 cm
Radius of base of the cone,
\[r = \frac{14}{2} = 7 cm\]
Height of the cone, h = 14 cm
Slant height of the cone, \[l = \sqrt{h^2 + r^2}\]
\[\Rightarrow l = \sqrt{\left( 7 \right)^2 + \left( 14 \right)^2}\]
\[ \Rightarrow l = \sqrt{49 + 196} = \sqrt{245} = 7\sqrt{5} cm\]
Surface area of the cone
\[= \pi r\left( r + l \right)\]
\[ = \pi\left( 7 \right)\left( 7 + 7\sqrt{5} \right)\]
\[ = 154\left( 1 + \sqrt{5} \right) {cm}^2\]
Surface area of the remaining solid = Surface area of the cube − surface area of the cone
\[= 6 a^2 - \frac{1}{3} \pi r^2 h\]
\[ = 6 \times \left( 14 \right)^2 - \left[ 154\left( 1 + \sqrt{5} \right) \right]\]
\[ = 1022 + 154\sqrt{5} {cm}^2\]
APPEARS IN
RELATED QUESTIONS
A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass. [use π=22/7]
A 5 m wide cloth is used to make a conical tent of base diameter 14 m and height 24 m. Find the cost of cloth used at the rate of Rs 25 per metre ?\[[Use \pi = \frac{22}{7}]\]
The radii of ends of a frustum are 14 cm and 6 cm respectively and its height is 6 cm. Find its volume \[\pi\] = 3.14)
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.
A solid cone of base radius 10 cm is cut into two part through the mid-point of its height, by a plane parallel to its base. Find the ratio in the volumes of two parts of the cone.
A bucket, made of metal sheet, is in the form of a cone whose height is 35 cm and radii of circular ends are 30 cm and 12 cm. How many litres of milk it contains if it is full to the brim? If the milk is sold at Rs 40 per litre, find the amount received by the person.
A metallic hemisphere is melted and recast in the shape of a cone with the same base radius R as that of the hemisphere. If H is the height of the cone, then write the values of \[\frac{H}{R} .\]
A cylinder and a cone are of the same base radius and of same height. Find the ratio of the value of the cylinder to that of the cone.
The diameters of the ends of a frustum of a cone are 32 cm and 20 cm. If its slant height is 10 cm, then its lateral surface area is
A solid metallic right circular cone 20 cm high and whose vertical angle is 60°, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter `1/12` cm, then find the length of the wire.
