Planter Box Soil Order
The horticulture class is building two raised planters for the courtyard. Planter A is a cube with volume 0.5 m³. Planter B is a rectangular prism measuring 1.2 m by 0.5 m by 0.8 m. Soil is sold only in 40-liter bags, and the class must buy whole bags. A delivery doorway is 0.8 m wide, so each planter's square footprint must fit through it; Planter A's footprint side length equals the cube's edge. The class has one cart that can carry at most 500 kg, and each bag of dry soil weighs 18 kg. The teacher claims Planter A will need exactly one more bag than Planter B and that Planter A's edge length is a rational number of meters. A student argues the cube cannot fit through the doorway because its edge is longer than 0.8 m. Use 1 m³ = 1,000 L.
1. Convert both volumes to liters, determine the number of bags needed for each planter, and evaluate the teacher's 'one more bag' claim.
Read the worked answer
Planter B: 1.2 × 0.5 × 0.8 = 0.48 m³. In liters, A is 0.5 × 1000 = 500 L and B is 0.48 × 1000 = 480 L. Bags are whole 40-L bags, so A needs 500 ÷ 40 = 12.5, which rounds up to 13 bags; B needs 480 ÷ 40 = 12 bags. The difference is 13 − 12 = 1 bag, so the teacher's claim is correct. Rounding down would leave A short 20 L of soil, so rounding up is required.
2. Determine whether Planter A's edge length is rational or irrational, give it to the nearest centimeter, and decide whether the student's doorway claim is correct.
Read the worked answer
The edge length e satisfies e³ = 0.5, so e = cube root of 0.5 ≈ 0.7937 m. This is irrational: if e = p/q in lowest terms, then p³/q³ = 1/2, so 2p³ = q³. Then q³ is even, so q is even; writing q = 2k gives 2p³ = 8k³, so p³ = 4k³, which makes p even. That contradicts p/q being in lowest terms. Thus e ≈ 0.794 m = 79.4 cm, which rounds to 79 cm. Since the doorway is 80 cm wide, the cube fits with about 0.6 cm clearance, so the student is wrong.
3. Can all 25 bags be moved in one cart trip? Show the weight calculation and state any practical assumption.
Read the worked answer
Total bags = 13 + 12 = 25. Weight = 25 × 18 = 450 kg, which is under the 500-kg limit, so one trip is possible with 50 kg to spare. This assumes the cart's limit includes the soil bags and that the bags can be stacked safely. If the cart's own weight or a passenger counts toward the 500-kg limit, the answer could change.