Choosing a Garden Path Design Under a Budget
A community group is building a rectangular garden that is 18 meters long and 12 meters wide. A path 1.5 meters wide will run inside the garden along all four sides, and the remaining rectangle will be planted with soil. A fence costs $12 per meter and must surround the outer edge of the garden. Soil costs $8 per square meter of planted area. A gate costs $150. The group has a budget of $2,000. One member suggests moving the path outside the garden instead, so the planted rectangle stays 18 by 12 meters and the path surrounds it.
1. Compute the total cost of the plan with the path inside the garden.
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With a 1.5 m path inside on all four sides, the planted rectangle shrinks by 3 m in each dimension: length 18 − 2(1.5) = 15 m, width 12 − 2(1.5) = 9 m. Planted area = 15 × 9 = 135 m². Soil cost = 135 × $8 = $1,080. The fence still surrounds the original 18 by 12 rectangle: perimeter = 2(18 + 12) = 60 m, so fence cost = 60 × $12 = $720. Adding the gate: $1,080 + $720 + $150 = $1,950. This is under the $2,000 budget by $50.
2. Compute the cost if the path is moved outside the garden, and decide whether the group can afford that plan.
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If the path moves outside, the planted rectangle remains 18 by 12 = 216 m², so soil costs 216 × $8 = $1,728. The fence must now enclose the path, so the outer dimensions grow by 1.5 m on each side: length 18 + 3 = 21 m, width 12 + 3 = 15 m. Perimeter = 2(21 + 15) = 72 m, so fence cost = 72 × $12 = $864. Total with gate: $1,728 + $864 + $150 = $2,742. That exceeds the $2,000 budget by $742, so the group cannot afford the outside-path plan.
3. Which design should the group choose, and what trade-off does the choice involve?
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The group should choose the inside-path design because it costs $1,950, within the $2,000 budget, while the outside-path design costs $2,742. The trade-off is that the inside path reduces the planted area from 216 m² to 135 m², so the garden has less growing space. But the outside design saves that planted area only by making the fence perimeter larger and the total cost unaffordable. Since the budget is a hard limit, the inside design is the workable choice.