From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and of base radius 6 cm is hollowed out. Find the volume of the remaining solid. Also, find the total surface area of the remaining solid. [Take π = 3.14.]
Height of solid cylinder = h = 8 cm
Radius of solid cylinder = r = 6 cm
Volume of solid cylinder = πr2h
= 3.14 × 6 × 6 × 8 cm3
= 904.32 cm3
Curved Surface area of solid cylinder = 2πrh
Height of conical cavity = h = 8 cm
Radius conical cavity = r = 6 cm
Volume of conical cavity = 1/3 πr2h
= 1/3 × 3.14 × 6 × 6 × 8 cm3
= 301.44 cm3
Let l be the slant height of conical cavity
l2 = r2 + h2
⇒ l2 = (62 + 82) cm2
⇒ l2 = (36 + 64) cm2
⇒ l2 = 100 cm2
⇒ l = 10 cm
Curved Surface area of conical cavity = πrl
Since, conical cavity is hollowed out from solid cylinder, so, volume and total surface area of remaining solid will be found out by subtracting volume and total surface area of conical cavity from volume and total surface area of solid cylinder.
Volume of remaining solid = Volume of solid cylinder – Volume of conical cavity
Volume of remaining solid = 904.32 cm3 – 301.44 cm3
= 602.88 cm3
Total surface area of remaining solid = Curved Surface area of solid cylinder + Curved Surface area of conical cavity + Area of circular base
Total surface area of remaining solid = 2πrh + πrl + πr2
= πr × (2h + l + r)
= 3.14 × 6 × (2 × 8 + 10 + 6) cm2
= 3.14 × 6 × 32 cm2
= 602.88 cm2