Raised Planter Boxes: Removing a Corner and Filling Below the Rim
A community garden committee is building two raised planter boxes from frames that are 1.5 feet tall. Box 1 has a rectangular base 6 feet by 4 feet. Box 2 has an L-shaped base: a 6-foot by 4-foot rectangle with a 3-foot by 2-foot rectangle removed from one corner. The committee buys soil in bags holding 2 cubic feet each, and the supplier says every box should be filled to 3 inches below the top rim so plants have room and rain does not overflow. A volunteer computes Box 2's soil volume as (6 × 4) × 1.5 − (3 × 2) = 36 − 6 = 30 cubic feet and plans to order 15 bags for that box alone. A second volunteer wants to fill both boxes all the way to the rim because "the extra soil is free space we already paid for." The committee needs one accurate bag count for both boxes together.
1. Find the volunteer's error in the Box 2 volume of 30 cubic feet, give the correct volume of Box 2, and explain why the removed corner must be handled the way you describe.
Read the worked answer
The volunteer subtracted the area of the missing corner (6 square feet) as if it were a volume, forgetting that the missing corner also has height.
Correct method: subtract the missing prism's volume. (6 × 4 × 1.5) − (3 × 2 × 1.5) = 36 − 9 = 27 cubic feet.
Equivalently, find the base area first, then multiply by height: (24 − 6) × 1.5 = 18 × 1.5 = 27 cubic feet.
Both routes give 27 because volume is base area times height, and the L-shaped base has area 18 square feet throughout the full 1.5-foot height. Subtracting 6 instead of 9 undercounts the soil needed by 3 cubic feet - one and a half bags - which would leave the box noticeably short.
2. Using the supplier's 3-inches-below-the-rim rule, compute the total soil needed for both boxes and the number of 2-cubic-foot bags to buy. Justify the rounding decision.
Read the worked answer
Convert the height: 3 inches = 0.25 foot, so usable height = 1.5 − 0.25 = 1.25 feet.
Box 1 (rectangular): 6 × 4 = 24 square feet of base; 24 × 1.25 = 30 cubic feet. Box 2 (L-shaped): 18 square feet of base; 18 × 1.25 = 22.5 cubic feet.
Total: 30 + 22.5 = 52.5 cubic feet.
Bags: 52.5 ÷ 2 = 26.25. Buying 26 bags supplies only 52 cubic feet, which is 0.5 cubic foot short, so the count must round up: 27 bags.
Rounding down is not a safe convenience here because soil cannot be bought in fractions of a bag and a half-filled box cannot be topped off from nothing. Filling to the rim instead would require 36 + 27 = 63 cubic feet, or 32 bags, wasting soil and removing the overflow space the supplier recommended.
3. Bags cost $8.50 each and the committee has $200. Determine whether the 27-bag plan fits the budget, and describe one realistic adjustment if it does not.
Read the worked answer
27 × $8.50 = 27 × 8 + 27 × 0.50 = 216 + 13.50 = $229.50. That is $29.50 over the $200 budget.
Possible adjustments, each of which must be checked against the 52.5-cubic-foot requirement:
- Raise more money: $29.50 more, and a purchase of 28 bags would add a small cushion ($238) in case a bag is spilled or a box settles.
- Reduce the fill depth: the committee could stop at 5 inches below the rim instead of 3. Usable height becomes 1.333... feet, giving roughly 24 × 1.333 + 18 × 1.333 ≈ 32 + 24 = 56 - still too much, so this alone does not work; the depth change saves only about 4.5 cubic feet (about 2 bags, $17).
- Shrink Box 2 or use the removed corner space as a separate small bed is not a savings, since the soil is still needed.
Buying 23 bags ($195.50) fits the budget but supplies only 46 cubic feet, well under the 52.5 needed, so the budget cannot be met by cutting bags alone.