Grain silo: volume, paint coverage and storage capacity
A secondary school is installing a grain silo made from a metal cylinder with a cone on top. The cylinder has radius 1.4 m and height 3.5 m. The conical roof has vertical height 0.9 m and sits exactly on the cylinder. The silo will be painted on the outside: the curved surface of the cylinder and the curved surface of the cone. The flat base is not painted. Paint is sold in 5-litre drums. Each drum covers 15 m² per coat and costs MWK 18,500. Two coats are required. A 10% allowance for wastage must be added to the total painted area. The school also wants to know how much grain the silo can hold when filled to 90% of its capacity, and whether six drums of paint are enough. The bursar is worried about overspending and wants the number of drums justified, not guessed.
1. Calculate the volume of the silo, correct to 2 decimal places. Show the cylinder and cone parts separately.
Read the worked answer
Decompose the silo into a cylinder and a cone.
Cylinder: V = πr²h = π × 1.4² × 3.5 = 6.86π = 21.551 m³.
Cone: V = (1/3)πr²H = (1/3) × π × 1.4² × 0.9 = 0.588π = 1.847 m³.
Total = 21.551 + 1.847 = 23.398 ≈ 23.40 m³ (2 d.p.).
This ignores wall thickness, so it is the internal capacity.
2. Calculate the total area to be painted, including the 10% wastage and two coats. Determine how many drums are needed and the paint cost.
Read the worked answer
Curved cylinder area = 2πrh = 2π × 1.4 × 3.5 = 9.8π = 30.788 m².
Cone slant height l = √(1.4² + 0.9²) = √2.77 = 1.6643 m.
Curved cone area = πrl = π × 1.4 × 1.6643 = 7.320 m².
Painted area before waste = 30.788 + 7.320 = 38.108 m².
Add 10%: 38.108 × 1.10 = 41.919 m².
Two coats: 41.919 × 2 = 83.838 m².
Drums needed = 83.838 ÷ 15 = 5.589, so 6 drums.
Cost = 6 × 18,500 = MWK 111,000.
Six drums are enough; five drums cover only 75 m², which is short of 83.8 m².
3. If the silo is filled to 90% of its capacity, how many full 20-litre buckets can be filled?
Read the worked answer
90% of the volume = 0.90 × 23.398 = 21.058 m³.
Convert to litres: 1 m³ = 1,000 litres, so 21,058 litres.
Number of full 20-litre buckets = 21,058 ÷ 20 = 1,052.9.
So 1,052 full buckets can be filled. The remaining 18 litres is not a full bucket.