Greenhouse Glazing and Ridge Height
A greenhouse sits on a 6 m by 10 m rectangular footprint. The vertical side walls are 2.4 m high. The roof is a symmetrical gable whose ridge runs parallel to the 10 m sides and sits 1.5 m above the top of the walls, so each of the two roof slopes covers half of the 6 m width. The glazing contractor will cover the two roof slopes, the two triangular gable ends, and the two long side walls; the floor is not glazed. A 1.0 m by 2.0 m personnel door in one gable end is excluded. Glazing panels measure 1.2 m by 2.4 m, and the contractor orders 10% extra panels for cuts and breakage. The owner is considering raising the ridge to 1.8 m above the walls and asks whether that change would add exactly one panel to the order and whether it would increase the internal volume by more than 10%.
1. Calculate the internal volume at the current ridge height, and at the proposed ridge height. Does the proposed change increase volume by more than 10%?
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The interior is a rectangular prism plus a triangular prism. Rectangular volume (=6\times10\times2.4=144) m³. Triangular cross-section area (=\frac12\times6\times1.5=4.5) m²; over 10 m length, volume (=45) m³. Total (=189) m³. With ridge 1.8 m, triangle area (=\frac12\times6\times1.8=5.4) m², volume 54 m³, total 198 m³. Increase (=(198-189)/189=9/189=0.0476), about 4.8%, not more than 10%. So the volume claim is false.
2. Calculate the glazed area and the number of panels to order at the current ridge height.
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Roof slope length (=\sqrt{3^2+1.5^2}=\sqrt{11.25}=3.354) m. Two roof slopes: (2\times10\times3.354=67.08) m². Each gable end area (=6\times2.4+\frac12\times6\times1.5=14.4+4.5=18.9) m²; two ends (=37.8) m². Side walls (=2\times10\times2.4=48) m². Gross glazing (=67.08+37.8+48=152.88) m². Subtract door (1.0\times2.0=2.0) m², giving 150.88 m². Add 10%: (150.88\times1.10=165.97) m². Each panel is (1.2\times2.4=2.88) m². Panels (=165.97/2.88=57.63), so order 58 panels after rounding up.
3. At the proposed ridge height, recalculate the panel order. Does it add exactly one panel? Combine this with the volume result to give the owner a corrected recommendation.
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New slope length (=\sqrt{3^2+1.8^2}=\sqrt{12.24}=3.499) m. Roof area (=2\times10\times3.499=69.97) m². Each gable (=14.4+\frac12\times6\times1.8=14.4+5.4=19.8) m²; two (=39.6). Total gross (=69.97+39.6+48=157.57); less door 2.0 =155.57 m²; plus 10% =171.13 m². Panels (=171.13/2.88=59.42), so 60 panels. This is two more than the 58-panel order, not one. The volume increase is 4.8%, not more than 10%. The owner should be told both corrections: the ridge change requires two extra panels and adds under 5% volume.