Shamiana Canvas and Air Space: Two Different Limits
For the annual function a school orders a shamiana: a cylindrical wall 2.1 m high and 3 m in radius, topped by a conical roof of slant height 5 m. The canvas comes on a roll and is billed at ₹70 per square metre; the tailor adds 5% extra for seams and wastage, and canvas can be bought only in whole square metres. The municipal fire officer insists on at least 1.5 m³ of air space per person, and the school has invited 70 guests. The tailor's quotation is for 100 m² of canvas; the school has sanctioned ₹6,500. The head teacher also wants to know how many guests can legally be seated if the invitation list has to be trimmed. Nothing may be counted twice - the roof sits on the wall, so the circular base of the cone is never stitched to anything - and the floor is open ground, not canvas.
1. Calculate the canvas actually required (including the tailor's 5%) and the cost at ₹70 per m². Does the order fit the ₹6,500 sanction?
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Curved surface of the cylindrical wall = 2πrh = 2 × (22/7) × 3 × 2.1 = 39.60 m². Curved surface of the conical roof = πrl = (22/7) × 3 × 5 = 47.14 m². The floor is open ground and the circle where roof meets wall is internal - neither is stitched, so neither is added.
Net canvas = 39.60 + 47.14 = 86.74 m². With 5% for seams and wastage: 86.74 × 1.05 = 91.08 m² → buy 92 m² (whole square metres only).
Cost = 92 × 70 = ₹6,440, which is ₹60 inside the ₹6,500 sanction. So the job is affordable - but only if 92 m² is ordered.
2. Compute the air space inside the shamiana and decide how many of the 70 invited guests may legally be seated.
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Cone height: H = √(l² − r²) = √(25 − 9) = 4 m.
Cylinder volume = πr²h = (22/7) × 9 × 2.1 = 59.40 m³. Cone volume = (1/3)πr²H = (1/3) × (22/7) × 9 × 4 = 37.71 m³. Total air space = 59.40 + 37.71 = 97.11 m³.
Guests allowed = 97.11 ÷ 1.5 = 64.74. Since a person cannot be split, round down: 64 guests.
With 70 guests the requirement would be 105 m³, about 8 m³ more than the shamiana holds. The invitation list must be trimmed by at least 6 people, and the fire officer's figure - not the school's - governs.
3. The tailor's quotation is for 100 m². Rule on it: is it adequate, wasteful, or unaffordable?
Read the worked answer
Adequacy: 100 m² exceeds the 91.08 m² required, so the shamiana could certainly be cut from it - there is no shortage of material.
Waste: 100 − 92 = 8 m² unused, worth 8 × ₹70 = ₹560.
Affordability: 100 × 70 = ₹7,000, which is ₹500 over the ₹6,500 sanction.
Judgement: the quotation is adequate but both wasteful and unaffordable. The correct instruction is to bill 92 m² (₹6,440), provided the roll can be cut to that length; if the supplier sells only full 10 m² lengths the school must either raise the sanction to ₹7,000 or reduce the wastage allowance. Note that trimming the canvas is not an option for the fire limit - the two limits (material cost and air space) are independent, and satisfying one does not satisfy the other.