Mulch, Borders, and a Break-Even Point
A school is refreshing the safety surface around a play structure measuring 8 m by 6 m. The structure sits in the middle of a gravel area, and rules require wood mulch in a 1 m wide strip all the way around it; the ground directly under the structure is not mulched. Two suppliers quote:
- Supplier A: $200 delivery, plus $7 per square metre.
- Supplier B: $40 delivery, plus $11 per square metre.
A student submits this work: « Area = 8 × 6 = 48 m². Cost A = 200 + 7(48) = $536. Cost B = 40 + 11(48) = $568. Supplier A is cheaper. »
The teacher confirms the multiplication and addition are correct but says the recommendation is wrong.
1. Find the error in the area calculation, then state how the recommendation changes.
Read the worked answer
The error: the student mulched the rectangle under the structure, which the rules exclude. The mulch covers only the 1 m border.
- Outer rectangle: (8 + 1 + 1) × (6 + 1 + 1) = 10 × 8 = 80 m²
- Inner rectangle (under structure): 8 × 6 = 48 m²
- Border only: 80 − 48 = 32 m²
Equivalently: two 10 × 1 strips plus two 6 × 1 strips = 20 + 12 = 32 m².
Revised costs:
- A: 200 + 7(32) = $424
- B: 40 + 11(32) = $392
Supplier B is cheaper by $32 - the opposite of the student's conclusion. The arithmetic was fine; the model of the situation was wrong, which is the more serious error.
2. The committee widens the strip to 1.5 m. Recompute and recommend, showing that your answer is consistent with a general rule.
Read the worked answer
- Outer: (8 + 3) × (6 + 3) = 11 × 9 = 99 m²
- Inner: 48 m²
- Strip: 99 − 48 = 51 m²
Costs:
- A: 200 + 7(51) = 200 + 357 = $557
- B: 40 + 11(51) = 40 + 561 = $601
Supplier A is cheaper by $44.
General rule: set 200 + 7x = 40 + 11x → 160 = 4x → x = 40 m². Below 40 m², B wins; above 40 m², A wins. At 32 m² B wins (consistent), at 51 m² A wins (consistent). The check at x = 40: both equal $480.
3. A committee member argues: « Just use the break-even point - above 40 m², use B, since B has the lower rate per square metre. » Identify the flaw in this reasoning.
Read the worked answer
The claim reverses the rule and misreads what a rate means.
- A's rate is $7/m², B's is $11/m². A has the lower variable cost; B only looks cheaper at small jobs because A charges $160 more up front in delivery.
- As area grows, A's fixed disadvantage is spread thinner and eventually disappears. So above 40 m², A is cheaper; below 40 m², B is cheaper.
The member has confused "lower rate" with "lower total." A lower per-unit rate always wins at sufficiently large quantities when the fixed costs can be recovered - that is exactly what the break-even point measures. The correct statement is: below 40 m² use B, above 40 m² use A, and at exactly 40 m² either costs $480.