Raised-Bed Design Trade-off
Ms. Okonkwo's Grade 7 class is building one raised garden bed for the courtyard. Design A is a rectangular prism 4 m long, 1.5 m wide, and 0.4 m deep. Design B is a cylinder with diameter 3 m and depth 0.4 m. Edging for the perimeter costs $8.50 per metre. Soil comes in 0.6 m³ bags at $12 each and cannot be bought in partial bags. The class has $150. The garden club requires at least 2.5 m³ of soil volume for the root vegetables. The class wants the design that meets the soil requirement and costs the least, using π ≈ 3.14.
1. Calculate the soil volume and the number of 0.6 m³ bags needed for each design. Then calculate the total cost (edging plus soil) using π ≈ 3.14.
Read the worked answer
Design A: V = 4 × 1.5 × 0.4 = 2.4 m³. Bags = 2.4 ÷ 0.6 = 4 exactly. Soil cost = 4 × $12 = $48. Perimeter = 2(4 + 1.5) = 11 m. Edging cost = 11 × $8.50 = $93.50. Total = $93.50 + $48 = $141.50.
Design B: radius = 3 ÷ 2 = 1.5 m. V = πr²h ≈ 3.14 × 1.5² × 0.4 = 3.14 × 2.25 × 0.4 = 2.826 m³. Bags = 2.826 ÷ 0.6 = 4.71, so 5 bags. Soil cost = 5 × $12 = $60. Circumference = πd ≈ 3.14 × 3 = 9.42 m. Edging cost = 9.42 × $8.50 = $80.07. Total = $80.07 + $60 = $140.07.
Both totals are under $150.
2. Which design meets the garden club's 2.5 m³ minimum? Which design should the class choose if it wants the lowest total cost? Address the conflict.
Read the worked answer
Design A has only 2.4 m³, which is below 2.5 m³, so it fails the requirement even though it is close. Design B has 2.826 m³, so it meets the requirement. Design B is also slightly cheaper: $140.07 versus $141.50. The cylinder uses one extra bag of soil ($12 more), but its circumference is 9.42 m instead of 11 m, saving $13.43 on edging; the net saving is about $1.43. Choose Design B: it meets the soil minimum and costs less.
3. If edging costs $10.00 per metre instead of $8.50, what is the maximum edging price per metre that would let Design B stay within the $150 budget? Use the same soil cost.
Read the worked answer
Design B's soil cost is fixed at $60 for 5 bags. Within a $150 budget, at most $150 − $60 = $90 can be spent on edging. The circumference is about 9.42 m, so the maximum price per metre is 90 ÷ 9.42 ≈ $9.55/m. At $10/m, edging would cost $94.20 and the total would be $154.20, which is $4.20 over budget. The class would need edging at about $9.55/m or less, or another cost reduction.