Question

A tent is in the form of a right circular cylinder surmounted by a cone. The diameter of cylinder is 24 m. The height of the cylindrical portion is 11 m while the vertex of the cone is 16 m above the ground. Find the area of the canvas required for the tent.

Answer 1

The tent being in the form of a cone surmounted on a cylinder the total amount of canvas required would be equal to the sum of the curved surface areas of the cone and the cylinder.

The diameter of the cylinder is given as 24 m. Hence its radius, r = 12 m. The height of the cylinder, h = 11 m.

The curved surface area of a cylinder with radius ‘r’ and height ‘h’ is given by the formula

Curved Surface Area of the cylinder =  2πrh

Substituting the values of r = 12 m and h = 11 m in the above equation

Curved Surface Area of the cylinder =  `2 (pi)(12)(11)`

=  `264pi`

The vertex of the cone is given to be 16 m above the ground and the cone is surmounted on a cylinder of height 11 m, hence the vertical height of the cone is h = 5 m. The radius of the cone is the same as the radius of the cylinder and so base radius, r = 12 m.

To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation

Slant height, l =  `sqrt(r^2 + h^2)`

= `sqrt(12^2 +5^2)`

= `sqrt(144+25`

=  `sqrt(169)`

l = 13 m

The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as

Curved Surface Area = ` pirl`

Substituting the values of r = 12 m and l = 13 m in the above equation

We get

Curved Surface Area of the cone = `pi (12)(13)`

= `156 pi`

Total curved surface area = Curved surface area of cone + curved surface area of cylinder

=  `156 pi + 264 pi `

= `420 pi `

=`((420)(22))/7`

= 1320

Thus the total area of canvas required is `1320  m^2`