📚 IGCSE WJEC Further Maths: Case Study Practical Exercise | IGCSE WJEC 进阶数学：案例分析实战演练
In IGCSE WJEC Further Mathematics, the ability to apply diverse techniques to a single real-world scenario is highly valued. This case study walks through the design of an open water trough, integrating algebra, trigonometry, differentiation, integration, matrices, and vectors. By following the step-by-step analysis, you will see how these topics interconnect and reinforce examination skills.
在 IGCSE WJEC 进阶数学中，将多种技巧应用于一个现实场景的能力备受重视。本案例分析将逐步展示一个开口水槽的设计过程，整合代数、三角、微分、积分、矩阵和向量。通过分步解析，你将看到这些主题如何相互关联，并增强应试技巧。
1. Case Background: Designing an Efficient Trough | 案例背景：设计高效水槽
A long metal sheet of width 90 cm is to be bent into an open trough with a trapezoidal cross-section. The sheet is creased along two lines parallel to its length, dividing the width into three equal parts of 30 cm each: a central base and two side flaps. By changing the angle θ that each side makes with the horizontal, we seek the maximum cross‑sectional area.
一块宽度为90 cm的长金属板将被弯曲成一个开口水槽，其截面为梯形。在板上沿长度方向压出两条折痕，将宽度三等分，每部分30 cm：一个中央底面和两个侧翼。通过改变每个侧翼与水平面的夹角 θ，我们希望获得最大的横截面积。
2. Deriving the Cross‑Sectional Area Function | 推导横截面积函数
top width equals b + 2 s cosθ. Hence the area is A = 1/2 (b + b + 2 s cosθ) × s sinθ = (b + s cosθ) s sinθ. Substituting b = 30, s = 30 gives A(θ) = (30 + 30 cosθ) × 30 sinθ = 900 (1 + cosθ) sinθ cm².
底面宽度 b = 30 cm，每个斜边长度 s = 30 cm。斜边的水平投影为 s cosθ，垂直高度为 s sinθ。上底宽度等于 b + 2 s cosθ。因此面积为 A = 1/2 (b + b + 2 s cosθ) × s sinθ = (b + s cosθ) s sinθ。代入 b=30, s=30 得 A(θ) = (30+30 cosθ)×30
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