Scale Drawing and Real-World Measurement Error
A landscaping class is designing a rectangular community garden on a plan with scale 1:150. On the plan the plot measures 84 mm by 56 mm. The path around the plot is to be 1.2 m wide on all sides. Fencing is sold in 8 m rolls and costs $46 per roll; a roll cannot be joined invisibly, so any part-roll counts as a full roll for pricing but offcuts can be used elsewhere. Topsoil for the beds (inside the path) is sold by the cubic metre and must be ordered to the nearest 0.5 m³, delivered at $58/m³. The class measures the plan with a ruler accurate to ±1 mm on each dimension.
1. Convert the plan dimensions to real dimensions, including the uncertainty, and find the fenced perimeter and number of fence rolls.
Read the worked answer
Real length = 84 mm × 150 = 12,600 mm = 12.6 m. Real width = 56 × 150 = 8,400 mm = 8.4 m. Uncertainty: ±1 mm on plan = ±150 mm = ±0.15 m per dimension. So length is 12.6 ± 0.15 m, width 8.4 ± 0.15 m.
The fence runs around the outer edge of the path. Path is 1.2 m wide, so outer dimensions = plot + 2 × 1.2 = length 12.6 + 2.4 = 15.0 m; width 8.4 + 2.4 = 10.8 m. Outer perimeter = 2(15.0 + 10.8) = 51.6 m. Rolls needed = 51.6 / 8 = 6.45 → 7 rolls. Cost = 7 × $46 = $322.
Check the uncertainty: worst case outer length 15.3 m, width 11.1 m, perimeter 52.8 m → still 7 rolls (52.8/8 = 6.6). Best case 50.4 m → 6.3, still 7 rolls. The rounding up to 7 is robust to the measurement error.
2. Calculate the topsoil volume for beds 0.25 m deep, and the delivered cost.
Read the worked answer
Beds are inside the path, i.e. the original plot: 12.6 m × 8.4 m = 105.84 m². Volume = 105.84 × 0.25 = 26.46 m³. Order to nearest 0.5 m³: 26.46 rounds to 26.5 m³. Cost = 26.5 × $58 = $1,537.
Now test the measurement uncertainty: smallest area = 12.45 × 8.25 = 102.71 m², volume 25.68 m³ → 25.5 m³; largest = 12.75 × 8.55 = 109.01 m², volume 27.25 m³ → 27.5 m³. So the order could range from 25.5 to 27.5 m³, a cost range of about $1,479 to $1,595. The judgement call: rounding to nearest 0.5 m³ hides a possible shortfall, so ordering 27 m³ (upper side) is safer than the central 26.5 m³ if running short is costly.
3. The class has a budget of $1,800 for fence and topsoil. Decide whether the plan is feasible and justify with a range.
Read the worked answer
Central estimate: fence $322 + topsoil $1,537 = $1,859, which exceeds $1,800 by $59. But the topsoil figure is uncertain. Using the low end (25.5 m³): 25.5 × 58 = $1,479; plus fence $322 = $1,801 - essentially at the budget. Using the high end (27.5 m³): $1,595 + $322 = $1,917, over budget.
So the plan is marginal: it is over budget on the central estimate and only just affordable at the most favourable measurement. A defensible recommendation is to reduce depth to 0.2 m (volume 21.17 m³ → 21 m³, cost $1,218; total $1,540, comfortably within budget) or to source cheaper topsoil. The judgement call is refusing to declare 'yes' or 'no' from a single central figure when the measurement uncertainty straddles the budget line.