The adjoining figure shows a wooden toy rocket which is in the shape of a circular cone mounted on a circular cylinder. The total height of the rocket is 26 cm, while the height of the conical part is 6 cm. The base of the conical portion has a diameter of 5 cm, while the base of the diameter of cylindrical portion is 3 cm. If the conical portion is to be painted green and the cylindrical portion is red, find the area of the rocket painted with each of these colours. Also find the volume of the wood in the rocket.
Use π = 3.14 and give answers correct to 2 decimal places.
Answers
(i) Area of the surface to be painted green = 51.025 cm²
(ii) Area of the surface to be painted red = 195.465 cm²
(iii) Total volume of wood in the rocket = 180.55 cm³
• Given,
Total height of the rocket = 26 cm
Height of the conical part (h) = 6 cm
Diameter of the base of conical portion = 5 cm
Diameter of the base of cylindrical portion = 3 cm
• Height of the cylindrical part (h') = Height of the rocket - Height of the conical portion = 26 cm - 6 cm = 20 cm
Radius of the base of conical portion (r) = 5 cm / 2 = 2.5 cm
Radius of the base of cylindrical portion (r') = 3 cm / 2 = 1.5 cm
• The rocket is painted on the outer surface.
• Surface area to be painted green = Curved surface area of the conical portion = πrl ( l is the slant height of the cone)
=> πrl = πr (√h²+r²)
= 3.14 × 2.5 cm (√36 + 6.25 cm²)
= 3.14 × 2.5 cm × √42.25 cm²
= 3.14 × 2.5 cm × 6.5 cm = 51.025 cm²
• Surface area to be painted red = Curved surface area of the cylindrical part + base
= 2πr'h' + πr'²
= πr' (2h' + r')
= 3.14 × 1.5 cm { (2 × 20 cm) + 1.5 cm }
= 3.14 × 1.5 cm (40 cm + 1.5 cm)
= 3.14 × 1.5 cm × 41.5 cm
= 195.465 cm²
• Volume of wood in the rocket= Volume of the conical portion + Volume of the cylindrical portion
• Volume of the conical portion = πr² × (h / 3)
=> Volume = 3.14 × (2.5 cm)² × (6 cm / 3)
= 3.14 × 6.25 cm² × 2 cm
= 39.25 cm³
• Volume of the cylindrical portion = πr'²h'
=> Volume = 3.14 × (1.5 cm)² × 20 cm
= 3.14 × 2.25 cm² × 20 cm
= 141.3 cm³
• Therefore, total volume of wood = 39.25 cm³ + 141.3 cm³ = 180.55 cm³