📚 Case Study Practice in Year 8 CAIE Further Mathematics | Year 8 CAIE 进阶数学：案例分析实战演练
Case studies are an essential part of the CAIE Further Mathematics curriculum for Year 8. They link abstract mathematical concepts to real-world scenarios, helping students develop problem-solving, modelling, and analytical thinking skills. In this article, we will work through several detailed case studies covering quadratic functions, geometric sequences, and trigonometry. Each case study includes a step-by-step breakdown, so you can see how to translate a written problem into equations, perform calculations, and interpret the results. Whether you are preparing for an exam or deepening your understanding, practising case studies will give you a solid foundation for IGCSE Additional Mathematics and beyond.
案例分析是Year 8 CAIE进阶数学课程的重要组成部分。它们将抽象的数学概念与现实世界的情景联系起来，帮助学生培养解决问题的能力、建模思维和分析技巧。本文将带领你完成几个详细的案例研究，涵盖二次函数、等比数列和三角学。每个案例都包含逐步拆解，让你看到如何将文字问题转化为方程、执行计算并解释结果。无论你是在准备考试还是深化理解，练习案例分析都将为IGCSE附加数学及更高阶的学习打下坚实基础。
1. Understanding Case Studies in Further Math | 理解进阶数学中的案例分析
A case study in Further Mathematics typically presents a situation described in words, often involving business, science, or everyday life. Your task is to identify which mathematical tools to apply – this could be functions, sequences, trigonometry, or coordinate geometry. You must then formulate a mathematical model, solve it accurately, and communicate your findings clearly. The challenge is not just in the computation, but in the translation from language to mathematics and back again.
进阶数学中的案例分析通常会给出一个用文字描述的情景，通常涉及商业、科学或日常生活。你的任务是识别需要采用哪种数学工具——可能是函数、数列、三角学或坐标几何。接着你必须建立数学模型、准确求解，并清晰地表达你的发现。挑战不仅在于计算，更在于从语言到数学、再从数学回到语言的转换。
2. Case Study 1: Maximising Revenue with Quadratic Functions | 案例一：利用二次函数最大化收入
e demand equation q = 200 - 4p. The weekly revenue R is calculated as price times quantity: R = p × q. The company wants to find the price that yields the maximum revenue.
一家小公司销售手工制品。市场调研表明，当单价为p美元时，每周销售数量q由需求方程 q = 200 - 4p 给出。每周收入R按价格乘以数量计算：R = p × q。公司希望找出能够产生最大收入的价格。
3. Breaking Down the Revenue Model | 分解收入模型
First, express R solely in terms of p. Substitute q = 200 - 4p into the revenue formula:
首先，用p单独表示R。将q = 200 - 4p代入收入公式：
R = p(200 - 4p) = 200p - 4p²
This is a quadratic function with a negative coefficient for p², meaning its graph is a downward-opening parabola. The maximum revenue occurs at the vertex of the parabola. For a quadratic in the form ax² + bx + c, the p-coordinate of the vertex is given by -b / (2a).
这是一个二次函数，p²的系数为负值，表明其图像是一条开口向下的抛物线。最大收入出现在抛物线的顶点处。对于形如 ax² + bx + c 的二次式，顶点的p坐标由 -b / (2a) 给出。
4. Solving the Quadratic Optimisation | 求解二次优化
Here, a = -4 and b = 200. So:
此处a = -4，b = 200。因此：
p = -200 / (2 × -4) = -200 / -8 = 25
The price that maximises revenue is $ = 25 back into the revenue equation: R = 200(25) - 4(25)² = 5000 - 4(625) = 5000 - 2500 = $2500. So the optimal price is $25, resulting in a weekly revenue of $2500.
使收入最大化的价格为25美元。为求出最大收入，将p = 25代回收入方程：R = 200(25) - 4(25)² = 5000 - 4(625) = 5000 - 2500 = $2500。因此最优价格为25美元，产生每周2500美元的收入。
5. Case Study 2: Geometric Sequences in Savings Plans | 案例二：储蓄计划中的等比数列
Emily decides to save money by depositing $80 each month into an account that pays an annual interest rate of 6% compounded monthly. She starts with an initial deposit of $80 at the end of the first month. She wants to know how many months it will take for the total amount saved, including interest, to reach at least $5000.
Emily决定每月存入80美元到一个年利率为6%、按月复利的账户。她在第一个月末首次存入80美元。她想知道经过多少个月，包括利息在内的总储蓄额能达到至少5000美元。
6. Deriving the Savings Formula | 推导储蓄公式
The monthly interest rate i = 6% ÷ 12 = 0.5% ture value of an ordinary annuity problem. After n months, the total amount A is given by:
月利率 i = 6% ÷ 12 = 0.5% = 0.005。这是一个普通年金的终值问题。经过n个月后，总金额A由下式给出：
A = 80 × [(1 + i)ⁿ - 1] / i
rm 80 and common ratio (1 + i). We need to find the smallest integer n such that A ≥ 5000.
该公式对首项为80、公比为(1 + i)的等比数列求和。我们需要找到最小的整数n，使A ≥ 5000。
7. Solving for the Required Time | 求解所需时间
Substituting i = 0.005:
代入i = 0.005：
80 × [(1.005)ⁿ - 1] / 0.005 ≥ 5000
Multiply both sides by 0.005 / 80:
两边乘以0.005 / 80：
(1.005)ⁿ - 1 ≥ (5000 × 0.005) / 80 = 25 / 80 = 0.3125
(1.005)ⁿ ≥ 1.3125
Taking natural logarithms (or base-10 logs) of both sides:
两边取自然对数（或常用对数）：
n ln(1.005) ≥ ln(1.3125)
n ≥ ln(1.3125) / ln(1.005) ≈ 0.27193 / 0.0049875 ≈ 54.52
r goal in the 55th month. It is always necessary to round up to the next integer since a partial month still requires a full deposit period.
因此n = 55个月。Emily将在第55个月达到目标。由于不足整月仍需完整的存款周期，因此总是需要向上取整。
8. Case Study 3: Trigonometric Height Calculation | 案例三：三角高程计算
A surveyor is measuring the height of a vertical tower. From point A on level ground, the angle of elevation to the top of the tower is 28°. From point B, which is 65 metres closer to the tower along the same straight line, the angle of elevation becomes 42°. Find the height of the tower.
一名测量员正在测量一座垂直塔的高度。从水平地面上的A点观测塔顶的仰角为28°。从B点出发，沿着同一直线向塔靠近65米后，仰角变为42°。求塔的高度。
9. Applying Trigonometric Ratios | 应用三角函数比
Let the foot of the tower be C, and its height be h. Let the distance from A to C be x metres. Then the distance from B to C is (x - 65) metres. From the right triangles formed:
设塔底为C，塔高为h。设A到C的距离为x米。则B到C的距离为(x - 65)米。由形成的直角三角形可得：
tan 28° = h / x
tan 42° = h / (x - 65)
These two equations give us a way to eliminate h or x.
这两个方程提供了一种消去h或x的方法。
10. Solving the System and Interpreting Results | 求解方程组并解释结果
From the first equation, h = x tan 28°. Substitute into the second:
由第一个方程得 h = x tan 28°。代入第二个方程：
x tan 28° = (x - 65) tan 42°
Expand and collect terms containing x:
展开并合并含x的项：
x tan 28° = x tan 42° - 65 tan 42°
x (tan 42° - tan 28°) = 65 tan 42°
x = 65 tan 42° / (tan 42° - tan 28°)
Using approximations tan 42° ≈ 0.9004 and tan 28° ≈ 0.5317:
使用近似值 tan 42° ≈ 0.9004，tan 28° ≈ 0.5317：
x ≈ 65 × 0.9004 / (0.9004 - 0.5317) = 58.526 / 0.3687 ≈ 158.74 m
wer is approximately 84.4 metres tall. In a real survey, you would use more precise values and consider measurement errors.
然后 h = 158.74 × 0.5317 ≈ 84.4 m。塔高约为84.4米。在实际测量中，你需要使用更精确的数值并考虑测量误差。
11. Common Pitfalls in Case Study Analysis | 案例分析常见误区
Many students rush to write equations without clearly defining variables. Always start by writing down what each symbol represents, with units. Another frequent mistake is forgetting to check whether the mathematical answer makes sense in context – a negative time or a distance longer than explained may indicate an algebraic error or a misapplied model. Rounding too early can lead to inaccurate final results, so keep full calculator precision during intermediate steps.
许多学生急于写出方程却没有清晰地定义变量。始终应该首先写下每个符号表示什么，并注明单位。另一个常见错误是忘记检查数学答案在语境下是否有意义——时间出现负值或距离超出说明的范围可能表明代数错误或模型使用不当。过早取整会导致最终结果不准确，因此在中间步骤应保留计算器的全部精度。
12. Final Tips for Case Study Success | 案例分析成功的最后建议
Practise deconstructing the problem statement by underlining key numbers and the unknown you need to find. Decide what branch of mathematics fits best, then build your model step by step. After solving, always re-read the question to ensure you have answered exactly what was asked. If time allows, verify your solution using a different method – for example, check the revenue maximum by testing values slightly above and below the optimum price. Consistent practice with varied scenarios will make you confident in tackling any case study in Year 8 CAIE Further Mathematics.
练习时要学会分解问题陈述，划出关键数字和需要求解的未知量。判断最适合的数学分支，然后逐步建立模型。求解后，务必重新阅读题目，确保你回答的正是题目所问。如果时间允许，用不同的方法验证你的解答——例如，通过测试略高于和略低于最优价格的值来检查收入最大值。在不同的情景中持续练习，将使你自信地应对Year 8 CAIE进阶数学中的任何案例分析。
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