A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of Rs. 500 per m² . (Note that the base of the tent will not be covered with canvas.)

As the capsule is in the shape of a cylinder with 2 hemispheres stuck to its ends, the diameter of the capsule is equal to the diameter of its cylindrical part and the diameter of its hemispherical part.

From Fig. 13.10, it’s clear that the capsule has the curved surface of two hemispheres and the curved surface of a cylinder.

Surface area of the capsule = 2 ×CSA of hemispherical part + CSA of cylindrical part

The surface area of the capsule can be found by using formulae;

CSA of the hemisphere = 2πr², where r is the radius of the hemisphere

CSA of the cylinder = 2πrh, where r and h are the radius and height of the cylinder respectively.

Diameter of the capsule, d = 5 mm

Radius of the hemisphere, r = d/2 = 5/2 mm

Radius of the cylinder, r = 5/2 mm

Length of the cylinder = Length of the capsule - 2 × radius of the hemisphere

h = 14 mm - 2 × 5/2 mm = 9 mm

Surface area of the capsule = 2 × CSA of hemispherical part +CSA of cylindrical part

= 2 × 2πr² + 2πrh

= 2πr (2r + h)

= [2 × 22/7 × 5/2 mm ×  (2 × 5/2 mm + 9 mm)]

= 110/7 mm × 14 mm

= 220 mm²

Thus, the surface area of the capsule is 220 mm².