📚 Case Study: Real-World Problem Solving | 案例分析实战演练
Case studies in mathematics bridge the gap between textbook theory and everyday life. They challenge you to read, interpret, and solve multi-step problems just like a professional would. In this article, we will work through a detailed scenario that integrates geometry, statistics, algebra, and financial maths – all within the Cambridge Year 10 curriculum.
数学案例分析将课本理论与日常生活联系起来，要求你像专业人士一样阅读、解释并解决多步骤问题。本文将带领你完成一个详细的实景演练，整合了几何、统计、代数和财务数学——全部属于剑桥 Year 10 课程范围。
1. Introducing the Garden Project | 引入花园项目
A local community group wants to transform an unused plot of land into a public garden. The shape of the plot is a rectangle measuring 20 metres by 15 metres, with a semicircular extension on one of the longer sides. Your task is to plan the garden, including pathways, planting areas, and the overall budget.
一个当地社区团体想将一块闲置土地改造成公共花园。地块形状为一个 20 米 × 15 米的矩形，并在其中一条长边外延伸出一个半圆形区域。你的任务是规划花园，包括步道、种植区域和整体预算。
The group has gathered some initial data: they have a list of preferred plants from a quick street survey, and they have set a rough spending limit of £800. You must use maths to turn these ideas into a practical and affordable design.
该团体已收集了一些初步数据：他们通过快速街头调查得到了一份受欢迎植物列表，并且设定了一个约 800 英镑的支出上限。你必须运用数学将这些想法转化为可行且不超支的设计。
2. Breaking Down the Problem | 分解问题
Before doing any calculations, read the brief carefully and identify the separate mathematical tasks. For this garden project, the key areas are: measuring the total land area, planning a gravel path around the edge, costing the path material, using survey data to choose plants, and checking the overall budget with an equation.
在进行任何计算之前，请仔细阅读任务简介，明确各项独立的数学任务。在这个花园项目中，关键任务包括：计算总土地面积、规划环绕边界的砾石步道、计算步道材料成本、利用调查数据选择植物，以及通过方程核对总体预算。
Breaking down a complex scenario into smaller, manageable steps is a skill that examiners value highly. Always list what you know and what you need to find out.
将复杂情景分解成更小、可操作的步骤是一项考官高度重视的技能。务必列出你已知的信息和需要求出的量。
3. Calculating the Area of the Garden | 计算花园面积
micircular part. The rectangle is 20 m long and 15 m wide, so its area = 20 × 15 = 300 m².
花园由一个矩形部分和一个半圆形部分组成。矩形长 20 米，宽 15 米，因此其面积 = 20 × 15 = 300 平方米。
= 7.5 m. The area of a full circle is πr², so the semicircle area = ½πr² = ½ × π × (7.5)².
半圆与矩形的 15 米边共用，因此其直径为 15 米，半径 r = 7.5 米。整个圆的面积 = πr²，所以半圆面积 = ½πr² = ½ × π × (7.5)²。
Semicircle area ≈ ½ × 3.1416 × 56.25 ≈ 88.36 m²
半圆面积 ≈ ½ × 3.1416 × 56.25 ≈ 88.36 平方米
Total garden area = 300 + 88.36 = 388.36 m². Always state the unit and round sensibly – here to two decimal places is appropriate.
花园总面积 = 300 + 88.36 = 388.36 平方米。务必标明单位并合理取整——这里保留两位小数是合适的。
4. Planning a Gravel Path | 规划砾石步道
A path 1.5 m wide will surround the entire garden. To find its area, we calculate the area of the enlarged outer shape and subtract the original garden area. The outer shape is also a rectangle plus a semicircle, but all dimensions are increased by the path width on each side.
一条 1.5 米宽的步道将环绕整个花园。为求出步道面积，我们先计算扩大后的外边界形状面积，再减去原花园面积。外边界形状同样由一个矩形和一个半圆组成，但所有尺寸都在各方向增加了步道宽度。
= 23 m and width becomes 15 + 2 × 1.5 = 18 m. So outer rectangle area = 23 × 18 = 414 m².
对于矩形部分，长度变为 20 + 2 × 1.5 = 23 米，宽度变为 15 + 2 × 1.5 = 18 米。因此外矩形面积 = 23 × 18 = 414 平方米。
th of 18 m, so its radius = 9 m. Outer semicircle area = ½π × 9² = ½ × π × 81 ≈ 127.23 m².
外部半圆的直径等于新的宽度 18 米，因此其半径 = 9 米。外半圆面积 = ½π × 9² = ½ × π × 81 ≈ 127.23 平方米。
Total outer area = 414 + 127.23 = 541.23 m². Path area = 541.23 – 388.36 = 152.87 m². This is the surface that needs gravel.
外边界总面积 = 414 + 127.23 = 541.23 平方米。步道面积 = 541.23 – 388.36 = 152.87 平方米。这就是需要铺设砾石的区域。
5. Budgeting for Materials | 材料预算
Gravel costs £12.50 per square metre, but we should add 10% 0.10 = 15.287 m². Total area to order = 152.87 + 15.287 = 168.157 m², rounded to 168.2 m².
砾石价格为每平方米 12.50 英镑，但我们应增加 10% 以考虑切割和损耗。净材料面积为 152.87 平方米，因此损耗余量 = 152.87 × 0.10 = 15.287 平方米。需订购的总面积 = 152.87 + 15.287 = 168.157 平方米，四舍五入为 168.2 平方米。
We can display the costs clearly in a table:
我们可以通过表格清晰地展示费用：
| Item | Area (m²) | Cost per m² | Subtotal |
|---|---|---|---|
| Net gravel area | 152.87 | £12.50 | £1910.88 |
| Waste (10%) | 15.29 | £12.50 | £191.13 |
| Total gravel cost | 168.16 | — | £2102.01 |
This initial cost is already far above the £800 budget, so we need to rethink – perhaps use a cheaper surface or reduce the path width. This is exactly the kind of reality check valued in case studies.
这一初步费用已远超 800 英镑预算，因此我们需要重新思考——或许使用更便宜的材料或减少步道宽度。这正是案例分析中重视的“现实检验”。
6. Choosing Plants with a Survey | 通过调查选择植物
A quick survey of 50 residents asked which type of flower they would most like to see. The results are shown in a frequency table.
一项对 50 位居民的快速调查询问了他们最希望看到哪种花卉。结果以频数表形式展示。
e = (15 ÷ 50) × 360° = 108°. Tulip: (10 ÷ 50) × 360° = 72°; Lavender: (8 ÷ 50) × 360° = 57.6°; Sunflower: (12 ÷ 50) × 360° = 86.4°; Daisy: (5 ÷ 50) × 360° = 36°.
玫瑰：15，郁金香：10，薰衣草：8，向日葵：12，雏菊：5。如果我们要绘制饼图，可以为每个扇区计算角度。玫瑰的角度 = (15 ÷ 50) × 360° = 108°；郁金香：(10 ÷ 50) × 360° = 72°；薰衣草：(8 ÷ 50) × 360° = 57.6°；向日葵：(12 ÷ 50) × 360° = 86.4°；雏菊：(5 ÷ 50) × 360° = 36°。
Always check that the angles sum to 360°: 108 + 72 + 57.6 + 86.4 + 36 = 360. Correct.
务必检查角度之和等于 360°：108 + 72 + 57.6 + 86.4 + 36 = 360。正确。
7. Using Statistics to Decide | 使用统计做决策
From the survey, the mode is Rose (15 votes), indicating that roses are the most popular choice. The median can also be found: listing the frequencies in order (5, 8, 10, 12, 15), the median is 10, corresponding to Tulips, but for categorical data the mode is often the most useful.
从调查数据看，众数是玫瑰（15 票），表明玫瑰是最受欢迎的选择。中位数也可以求出：将频数从小到大排列（5, 8, 10, 12, 15），中位数为 10，对应郁金香，但对于分类数据，众数通常最为有用。
nd daisies to respect all preferences. The exact areas will be set proportionally: Rose area = (15/50) × total planting area, and so on.
我们决定将最大的种植面积分配给玫瑰，其次为向日葵和郁金香，同时仍纳入较小的薰衣草和雏菊区域，以尊重所有偏好。具体面积将按比例分配：玫瑰面积 = (15/50) × 总种植面积，依此类推。
8. Constructing a Pie Chart | 构建饼图
In an exam, you may be asked to construct a pie chart accurately. For this data, you would draw a circle, measure each angle from the centre using a protractor, and label each sector clearly. Our calculated angles are:
在考试中，你可能会被要求准确地绘制饼图。对于这组数据，你需要画一个圆，用量角器从圆心量出每个角度，并清晰地标注每个扇区。我们计算出的角度如下：
| Flower | Frequency | Angle |
|---|---|---|
| Rose | 15 | 108° |
| Tulip | 10 | 72° |
| Lavender | 8 | 57.6° |
| Sunflower | 12 | 86.4° |
| Daisy | 5 | 36° |
Remember to write the category name and percentage (or frequency) on each sector. A key beside the chart also helps readability.
记得在每个扇区上标注类别名称和百分比（或频数）。图表旁的图例也有助于阅读。
9. Setting Up Equations for Plant Spacing | 建立植物间距方程
per plant. If we plant r roses and s sunflowers, the area equation is: 0.4r + 0.6s = 150.
假设我们分配 150 平方米用于花卉种植。每株玫瑰需要 0.4 平方米，每株向日葵需要 0.6 平方米。如果种植 r 株玫瑰和 s 株向日葵，面积方程为：0.4r + 0.6s = 150。
nce we cannot plant a fraction of a sunflower, we round down to s = 107, and then r = 214.
我们还希望玫瑰数量是向日葵的两倍，因此 r = 2s。代入得：0.4(2s) + 0.6s = 150 → 0.8s + 0.6s = 150 → 1.4s = 150 → s ≈ 107.14。由于不能种植零头向日葵，我们向下取整到 s = 107，则 r = 214。
Using simultaneous equations helped us balance space and design preferences.
利用联立方程帮助我们平衡了空间与设计偏好。
10. Final Cost Analysis | 最终成本分析
After realising the gravel cost was too high, we change the path material to wood chips at £4.80 per m², still requiring 152.87 m². Cost = 152.87 × 4.80 = £733.78, plus 10% waste = £733.78 × 0.10 = £73.38, total path cost = £807.16.
在意识到砾石费用过高后，我们将步道材料改为木屑，单价为每平方米 4.80 英镑，仍需 152.87 平方米。费用 = 152.87 × 4.80 = 733.78 英镑，加上 10% 损耗 = 73.38 英镑，步道总费用 = 807.16 英镑。
Roses cost £2.50 each and sunflowers £ ach. With 214 roses and 107 sunflowers, plant cost = 214 × 2.50 + 107 × 3.00 = 535 + 321 = £856.00. Adding a fixed labour charge of £100, total spending = 807.16 + 856.00 + 100 = £ till exceeds the budget, so we must reduce the number of plants or seek cheaper varieties.
玫瑰每株 2.50 英镑，向日葵每株 3.00 英镑。种植 214 株玫瑰和 107 株向日葵，植物费用 = 214 × 2.50 + 107 × 3.00 = 535 + 321 = 856.00 英镑。再加上固定人工费 100 英镑，总支出 = 807.16 + 856.00 + 100 = 1763.16 英镑。这仍超出预算，因此我们必须减少植株数量或寻找更便宜的品种。
gives a maximum of about 80 plants in total, which forces a complete redesign. This step mirrors the iterative nature of real projects.
预算约束可以用一个不等式表示：4.80 × (步道面积 × 1.1) + 2.50r + 3.00s + 100 ≤ 800。解这个不等式得出最多只能种植约 80 株，这迫使我们彻底重新设计。这一步骤反映了真实项目的迭代特性。
11. Exam Tips and Reflection | 考试技巧与反思
When tackling a case study in an exam, underline keywords, note all numerical data, and decide the order of calculations before you begin. If a cost exceeds the budget, as it often does on purpose, don't panic – just suggest a realistic alternative and show the corrected figures.
在考试中处理案例研究时，先划出关键词，记下所有数据，并在开始前确定计算顺序。如果费用超出预算（考试题常常故意如此设置），不要慌张——只需提出一个现实的替代方案并展示修正后的数据。
Always check units, round appropriately, and state your assumptions clearly. The mark scheme rewards method and clear reasoning, not just the final answer. Practising multi-topic scenarios like this garden project builds confidence for using maths beyond the textbook.
务必检查单位、适当取整并清晰陈述你的假设。评分标准不仅看重最终答案，更奖励解题方法和清晰的推理。多练习像花园项目这样融合多个专题的情景题，能帮助你建立运用数学解决实际问题的信心。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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