Similar Triangular Shade Sails Under Size and Budget Constraints
A community park is choosing between two triangular shade sails for a playground. Design A has side lengths 6 m, 8 m, and 10 m, with the 10 m side to be stretched horizontally between two posts. Design B is similar to Design A, but its shortest side is 9 m. The park has a horizontal cable that can hold a sail up to 12 m long, and the highest anchor point available is 6 m above the ground. For Design A, the installer will use the altitude to the 10 m side as the vertical support. Fabric costs $14 per square metre, and the park's fundraising budget for fabric is $700. The committee wants a sail that is as large as possible without exceeding the cable, height, or budget limits.
1. Verify that Design A is a right triangle, determine the scale factor for Design B, and list Design B's three side lengths.
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Design A: 6²+8²=36+64=100=10², so by the converse of the Pythagorean theorem it is a right triangle with the right angle between the 6 m and 8 m sides. Design B is similar with shortest side 9 m, so the scale factor is k=9/6=1.5. Its sides are 6(1.5)=9 m, 8(1.5)=12 m, and 10(1.5)=15 m.
2. Calculate the area and the altitude to the longest side for each design. Then determine whether Design B satisfies the cable and height constraints.
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Area of A = (1/2)(6)(8)=24 m². The altitude to the 10 m side satisfies Area=(1/2)(base)(height), so 24=(1/2)(10)h ⇒ h=48/10=4.8 m. For B, area scales by k²=2.25: 24(2.25)=54 m². Its altitude is 1.5(4.8)=7.2 m, or directly (9)(12)/15=7.2 m. Design B's longest side is 15 m, which exceeds the 12 m cable limit, and its altitude is 7.2 m, which exceeds the 6 m anchor-height limit. So B fails both constraints.
3. Recommend a design decision. Include the fabric cost for Design A and explain whether a scaled-down version of Design B could fit the budget and constraints. If not, what maximum scale factor relative to Design A satisfies all limits?
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Design A costs 24×14=$336, within the $700 budget, and its longest side is 10 m (≤12 m) and altitude 4.8 m (≤6 m), so it fits all constraints. Design B costs 54×14=$756, over budget as well as too large. If the park wants a larger similar sail, let k be the scale factor relative to A. Cable: 10k≤12 ⇒ k≤1.2. Height: 4.8k≤6 ⇒ k≤1.25. Budget: 24k²×14≤700 ⇒ 336k²≤700 ⇒ k²≤2.083 ⇒ k≤1.44. The tightest limit is k≤1.2, so the largest feasible similar sail has sides 7.2 m, 9.6 m, 12 m, area 24(1.2²)=34.56 m², and cost $483.84. Recommendation: choose Design A or the k=1.2 scaled version; do not choose Design B.