A toy is in the shape of a right circular cylinder with a hemisphere on one end a cone on the other. The radius and height of the cylindrical part are 5 cm and 13 cm respectively. The radii of the hemispherical and the conical parts are the same as that of the cylindrical part. Find the surface area of the toy, if the total height of the toy is 30 cm.
The toy is in the shape of a right circular cylinder surmounted by a cone at one end a hemisphere at the other.
Total height of toy = 30 cm
Height of cylinder = h = 13 cm
Radius of cylinder = r = 5 cm
Curved surface area of cylinder = 2πrh
= 2 × 22/7 × 5 × 13 cm2
Height of cone = h’ = Total height of toy – Height of cylinder – Radius of hemisphere
Height of cone = h’ = 30 cm – 13 cm – 5 cm = 12 cm
Radius of cone = r = Radius of cylinder
Radius of cone = r = 5 cm
Let the slant height of cone be l
l2 = h’2 + r2
⇒ l2 = 122 + 52 cm2 = 144 + 25 cm2 = 169 cm2
⇒ l = 13 cm
Curved surface area of cone = πrl
= 22/7 × 5 × 13 cm2
Radius of hemisphere = r = Radius of cylinder
Radius of hemisphere = r = 5 cm
Curved surface area of hemisphere = 2πr2
= 2 × 22/7 × 5 × 5 cm2
Surface area of the toy = Surface area of cylinder + Surface area of cone + Surface area of hemisphere
Surface area of toy = 2πrh + πrl + 2πr2
= πr (2h + l + 2r)
= 22/7 × 5 × (2 × 13 + 13 + 2 × 5) cm2
= 22/7 × 5 × 49 cm2
= 770 cm2
Surface area of toy is 770 cm2