Solar Panel on a Gable Roof
A homeowner is planning a solar installation on a gable roof. In the cross-section, the base of the roof triangle is 9.6 m wide. The left roof slope makes a 32° angle with the horizontal base, and the right slope makes a 41° angle. The ridge is directly above the base but not necessarily at its midpoint. The installer needs the ridge height and the two slope lengths before ordering rails. A local by-law requires solar rails to end at least 0.4 m from each edge of the roof slope for fire access. The homeowner wants to use a 5.8 m rail on the longer slope. A second quote assumes that increasing the base to 10.0 m while keeping both base angles would raise the ridge by exactly 0.4 m. The homeowner asks for a check of that assumption.
1. Determine the height of the ridge above the base and the lengths of the two roof slopes. Round to the nearest centimetre and show your trigonometric setup.
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Let A and B be the base endpoints, with angles 32° at A and 41° at B, and C the ridge. The third angle is C=180°-32°-41°=107°. By the sine law, AC/sin41° = 9.6/sin107°, so AC=9.6 sin41°/sin107°≈6.59 m. Similarly BC=9.6 sin32°/sin107°≈5.32 m. The height h from C to AB is h=AC sin32°≈6.59×0.5299≈3.49 m (or h=BC sin41°). To the nearest centimetre: slopes about 6.59 m and 5.32 m, height about 3.49 m.
2. The installer wants to place a 5.8 m solar rail on the longer slope, leaving at least 0.4 m clear at each end. Does it fit? Justify with calculations and state the maximum rail length that would fit.
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The longer slope is AC≈6.59 m. With 0.4 m clearance at each end, the maximum rail length is 6.59-0.8≈5.79 m. A 5.80 m rail is slightly too long: it exceeds the maximum by about 0.01 m, so it does not meet the by-law if the slope is exactly 6.59 m. Using more precise values, 6.586-0.8=5.786 m, so 5.80 m is still too long by about 1.4 cm. The installer should order a rail no longer than about 5.78 m, or shorten the required clearance.
3. A second installer claims that if the base were 10.0 m instead, keeping the same base angles, the height would increase by exactly 0.4 m. Evaluate this claim using proportional reasoning.
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If the base changes from 9.6 m to 10.0 m while both base angles stay the same, the new triangle is similar to the old one with scale factor 10.0/9.6≈1.0417. Every corresponding length, including the height, is multiplied by this factor. New height≈3.490×1.0417≈3.635 m. The increase is about 0.145 m, not 0.4 m. The claim is false; a 0.4 m increase would require a scale factor of about (3.490+0.4)/3.490≈1.115, i.e. a base of about 10.7 m, not 10.0 m.