Courtyard Shade Sail
A school is installing a triangular shade sail over a courtyard. The anchor points are A, B and C. The distance AB is 6.4 m, AC is 5.1 m, and the angle BAC is 72°. The fabric costs $38 per square metre and is sold only in whole square metres. An extra 8% of fabric area must be ordered for cutting waste. The three edges need binding that costs $12.50 per metre and is sold in 10 m rolls. At each of the three corners, an extra 0.35 m of binding is needed for fixing. The school has a budget of $850 for fabric and binding. The principal also wants to know whether a total of 20 m of binding is enough for the job. A supplier says the sail will need at least 17 m² of fabric before waste. The business manager must check the measurements and decide whether the budget and the total binding length are sufficient.
1. Calculate BC using the cosine rule, and the sail area using the sine formula. Give both to a sensible degree of accuracy.
Read the worked answer
Using the cosine rule:
$BC^2 = 6.4^2 + 5.1^2 - 2(6.4)(5.1)\cos 72^\circ$.
$\cos 72^\circ \approx 0.3090$, so
$BC^2 = 40.96 + 26.01 - 65.28(0.3090)$
$= 66.97 - 20.17 = 46.80$.
$BC \approx 6.84$ m.
Area $= \frac12(6.4)(5.1)\sin 72^\circ$
$= 16.32(0.9511) \approx 15.52$ m².
So $BC$ is about 6.84 m and the sail area is about 15.5 m².
2. Determine the fabric order and total fabric-plus-binding cost. Include 8% fabric waste, whole-square-metre fabric, and binding rolls. Decide whether the $850 budget is enough.
Read the worked answer
Fabric with waste: $15.52 \times 1.08 = 16.76$ m². Since fabric is sold in whole square metres, the school must order 17 m². Fabric cost is $17 \times 38 = $646.
Binding length needed is the perimeter plus corner allowances:
Perimeter $= 6.4 + 5.1 + 6.84 = 18.34$ m.
Corner allowance $= 3 \times 0.35 = 1.05$ m.
Total binding $= 18.34 + 1.05 = 19.39$ m.
Binding is sold in 10 m rolls, so two rolls are needed, giving 20 m. Binding cost is $20 \times 12.50 = $250.
Total cost $= 646 + 250 = $896. The budget is $850, so the school is short by $46. The budget is not enough. Also, the supplier's claim of at least 17 m² before waste is not supported: the actual area before waste is about 15.5 m², though 17 m² is still needed once waste and whole-square-metre ordering are included.
3. The principal asks whether a total of 20 m of binding is enough if the 0.35 m corner allowance is included. Check the claim and state any assumption.
Read the worked answer
Required binding is the perimeter plus three corner allowances:
$6.4 + 5.1 + 6.84 + 3(0.35) = 19.39$ m.
A total of 20 m is therefore enough, with about $20 - 19.39 = 0.61$ m spare. This assumes no extra cutting waste beyond the stated 0.35 m at each corner and that the binding follows the three straight edges. If there were additional overlap or cutting waste of more than about 3%, 20 m would not be sufficient.