Raised Garden Bed: Units That Do Not Convert Like Units
The seventh-grade garden club is building a raised bed in the courtyard. The bed is 8 feet long, 4 feet wide, and 10 inches deep. The plan calls for a 3-inch layer of gravel across the bottom for drainage, with soil filling the rest. Soil is sold in 1.5-cubic-foot bags and gravel in 0.5-cubic-foot bags. The club has $180; soil costs $7.25 per bag, gravel $4.50 per bag, and weed-barrier fabric for the bottom costs $1.10 per square foot. One student computes the volume as 8 × 4 × 10 = 320 cubic feet and asks for 214 bags of soil. A second student computes 96 × 48 × 10 = 46,080 and then divides by 12, getting 3,840 cubic feet.
1. Diagnose each student's error, then compute the correct soil volume and number of soil bags and gravel bags.
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Error 1: 8 × 4 × 10 multiplies two measurements in feet by one in inches. The product 320 is in mixed units and means nothing. The depth must be converted first: 10 in = 10/12 ft.
Error 2: converting cubic inches to cubic feet requires dividing by 12³ = 1,728, not by 12. Dividing by 12 is the linear conversion and overstates the volume by a factor of 144. (46,080 ÷ 1,728 = 26.67 ft³, which agrees with the correct method.)
Correct computation
- Footprint: 8 × 4 = 32 ft²
- Total interior volume: 32 × (10/12) = 26.67 ft³
- Gravel layer: 32 × (3/12) = 8 ft³ → 8 ÷ 0.5 = 16 bags
- Soil: 26.67 − 8 = 18.67 ft³ → 18.67 ÷ 1.5 = 12.44 → 13 bags (always round up on materials)
Note that 214 bags (from 320 ÷ 1.5) would be more than sixteen times the actual need.
2. Does the full plan fit the $180 budget? If not, propose one specific design change that brings it under budget and show the arithmetic for the revised plan.
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Original plan
- Soil: 13 × $7.25 = $94.25
- Gravel: 16 × $4.50 = $72.00
- Fabric: 32 ft² × $1.10 = $35.20
- Total = $201.45 → over budget by $21.45
Proposed change: reduce the bed depth from 10 inches to 8 inches, keeping the 3-inch gravel layer.
- Total volume: 32 × (8/12) = 21.33 ft³
- Gravel: still 32 × (3/12) = 8 ft³ → 16 bags = $72.00
- Soil: 21.33 − 8 = 13.33 ft³ → 13.33 ÷ 1.5 = 8.89 → 9 bags × $7.25 = $65.25
- Fabric: unchanged, $35.20
- New total = $172.45 → under budget with $7.55 to spare
(This is a judgment call: shallowing the bed reduces root depth. An alternative is to trim the footprint, but that also reduces the fabric and gravel costs, so it changes all three line items rather than one.)
3. A parent suggests doubling all three dimensions to "double the harvest." Evaluate the effect on materials, being explicit about which quantities scale by 2, by 4, and by 8.
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Doubling every linear dimension makes the new bed 16 ft × 8 ft × 20 in.
- Lengths (perimeter, lumber trim): ×2.
- Footprint / fabric: ×4 → 128 ft².
- Total volume: ×8 → 128 × (20/12) = 213.33 ft³.
- Gravel is different. The gravel layer is still only 3 inches thick, so gravel = 128 × 0.25 = 32 ft³, which is ×4 the original 8 ft³ - not ×8.
- Soil follows from subtraction: 213.33 − 32 = 181.33 ft³, versus the original 18.67 ft³ - about ×9.7.
The point: only pure geometric similarity scales rigidly. Because the gravel is specified by thickness rather than as a proportional fraction, it scales with area, and the soil absorbs the difference. "Double the size" is not a single operation, and the harvest claim itself is a separate assumption about plant yield per square foot, which would scale ×4 at best, not ×8.