Topsoil Order: Units, Waste Allowance, and a Budget Cap
A neighborhood association is refilling a raised bed that is a rectangular prism measuring 18 ft by 12 ft, to a depth of 6 in. The supplier sells bulk topsoil only in whole cubic yards at $32 per cubic yard, delivered, with a 3-cubic-yard minimum; bagged topsoil is sold individually in 1.5 ft³ bags at $4.25 each. Because the soil settles, the association's gardener insists on buying 10% more volume than the bed's geometric volume. A volunteer writes: 'Volume = 18 × 12 × 0.5 = 108 ft³, so 108 ÷ 27 = 4 yd³; I will order 4 cubic yards and we are done.' The treasurer has approved $150.
1. Check the volunteer's volume computation and determine the minimum purchase, in whole cubic yards and in bags, that satisfies the 10% settling allowance. Show the unit conversions.
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Geometric volume. Depth 6 in = 0.5 ft, so V = 18 × 12 × 0.5 = 108 ft³. Since 1 yd³ = 27 ft³, 108 ÷ 27 = 4 yd³. The volunteer's arithmetic is correct but stops too early.
Required volume. 108 × 1.10 = 118.8 ft³ = 118.8 ÷ 27 = 4.4 yd³.
Bulk. Sold in whole cubic yards, so 4 yd³ (108 ft³) is short of 118.8 ft³; the next whole yard is 5 yd³ = 135 ft³, costing 5 × $32 = $160.
Bags. 118.8 ÷ 1.5 = 79.2, so 80 bags are needed (rounding up, not down); 80 × $4.25 = $340.
The minimum purchase meeting the allowance is 5 yd³ of bulk soil.
2. The volunteer argues: 'Bagged soil must be cheaper per cubic foot, so 4 yd³ bulk plus a few bags will beat 5 yd³.' Evaluate this cost claim with a calculation.
Read the worked answer
Unit costs.
- Bulk: $32 ÷ 27 ft³ = $1.19 per ft³.
- Bags: $4.25 ÷ 1.5 ft³ = $2.83 per ft³.
Bags are more than twice as expensive per cubic foot, so the premise is false.
Mixed purchase. 4 yd³ supplies 108 ft³, leaving 118.8 − 108 = 10.8 ft³, which requires 10.8 ÷ 1.5 = 7.2, i.e. 8 bags. Cost = $128 + 8($4.25) = $128 + $34 = $162.
$162 > $160, so 5 yd³ of bulk is still the cheaper compliant option. The bag top-up would only win if the waste allowance were dropped, in which case 4 yd³ at $128 would exactly fill the bed.
3. The $150 cap cannot be met at the full depth. Determine the best feasible purchase under the cap, and state precisely how much the plan falls short of the 10% allowance.
Read the worked answer
Test the obvious options.
- 5 yd³ bulk = $160, over budget.
- 3 yd³ bulk = $96, leaving 118.8 − 81 = 37.8 ft³ = 25.2 → 26 bags = $110.50, total $206.50, far worse.
Best feasible mix. 4 yd³ bulk = $128, leaving $22. At $4.25 per bag that buys 5 bags ($21.25), total $149.25, under budget.
Actual volume. 108 + 5(1.5) = 115.5 ft³ against the 118.8 ft³ target: short by 3.3 ft³ (0.12 yd³).
Depth achieved. Bed area = 216 ft², so depth = 115.5 ÷ 216 = 0.535 ft ≈ 6.42 in, versus the 6.6 in the allowance calls for, a shortfall of about 0.18 in.
The honest recommendation is 4 yd³ plus 5 bags, with the shortfall disclosed, or a reduction of the bed area so the full 6.6 in fits the budget.