📚 Investigation 2: Stadium Seating | 探究二：体育场座位
Stadium seating design is a fascinating application of sequences and geometry in the IB Mathematics curriculum. In this investigation, we will model the heights of rows in a sports stadium to ensure that every spectator enjoys an unobstructed view of the playing field. By applying the concept of sightlines and deriving an arithmetic progression for row heights, we can determine the required rise per row and evaluate the overall structure.
体育场座位设计是IB数学课程中数列与几何的一个迷人应用。在本探究中，我们将对体育场中每排座位的高度进行建模，以确保每位观众都能无遮挡地看到比赛场地。通过应用视线概念并推导出排高度的等差数列，我们可以确定所需的每排升高量，并对整体结构进行评估。
1. The Sightline Problem | 视线问题
When spectators sit in a stadium, those in the rear rows must be able to see the focal point on the field over the heads of the people in front of them. If all rows were placed on a flat floor, the view would be blocked beyond the first few rows. The central challenge is to calculate the minimum vertical rise between consecutive rows so that the line of sight from each spectator to the focal point clears the head of the spectator immediately in front.
当观众坐在体育场中时，后排的观众必须能够越过前排观众的头顶看到场地上的焦点。如果所有排都设置在同一水平面上，那么只有前几排的视野不会被遮挡。核心挑战在于计算相邻排之间所需的最小垂直升高量，使得每位观众看向焦点的视线能刚好越过正前方观众的头顶。
2. Defining the Key Variables | 定义关键变量
To build a mathematical model, we must first define the essential quantities. Let the first row of seats be positioned at a horizontal distance D from the focal point F on the pitch. The horizontal spacing between successive rows is denoted by c. The eye height of a seated spectator in the first row is h₁. We introduce a sightline clearance C, which is the required vertical distance between the line of sight and the top of the head of the spectator in the row ahead. The head is assumed to extend a vertical distance t above eye level, but for simplicity many design guides combine t and the required clearance into a single C-value, typically 120 mm.
为了建立数学模型，我们必须首先定义基本量。设第一排座位位于距离球场焦点F水平距离为D的位置。相邻排之间的水平间距用c表示。第一排观众坐姿时的眼睛高度为h₁。我们引入视线间隙C，它表示视线与前一排观众头顶之间所需的垂直距离。通常假定头顶高出眼睛的垂直距离为t，但为了简化，许多设计指南将t和所需的间隙合并为一个C值，典型值为120 mm。
3. Geometric Derivation of Row Rise | 排升高的几何推导
mined. The focal point F is at (D, 0). For the spectator in the rear row, the line of sight to F must pass at least a distance C above the head of the front spectator. Using the straight‑ n and the condition that at x = 0 the line gives height h + C, we obtain the relationship:
考虑相邻的两排：前排眼睛坐标为(0, h)，后排眼睛坐标为(c, h+R)，其中R为待求的升高量。焦点F位于(D, 0)。对于后排观众，看向F的视线必须在前排观众头顶上方至少C处通过。根据直线方程以及在x=0处视线高度为h+C的条件，可以得到如下关系：
R = C − (h + C) · c / D
This formula shows that the required rise R decreases as the eye height h increases. Therefore, if we design the whole stand using the smallest eye height h₁, we will obtain a constant rise that works for the critical first row and gives even larger clearances for subsequent rows.
该公式表明，所需的升高量R会随着眼睛高度h的增加而减小。因此，如果我们使用最小的眼睛高度h₁来设计整个看台，将会得到一个恒定的升高量，它在最关键的第一排满足要求，而对后面的排则提供更大的间隙余量。
4. Arithmetic Sequence of Seat Heights | 座位高度的等差数列
rogression. Let the row number be n, with n = 1 for the first row. The eye height of the n‑th row is given by:
采用由h = h₁算出的恒定升高量R后，连续各排的眼睛高度便构成一个等差数列。设排数为n，第一排n=1。则第n排的眼睛高度为：
hₙ = h₁ + (n − 1) · R
This simple linear model is the foundation of many stadium cross‑sections. It makes construction straightforward because each step has the same height, and it allows architects to quickly estimate the total height of the stand.
这一简单的线性模型是许多体育场横截面设计的基础。由于每一步高度相同，施工变得简便，同时建筑师可以快速估算看台的总高度。
5. Worked Example with Typical Values | 典型取值计算示例
h₁ = 1.20 m, C = 0.12 m, c = 0.85 m, and D = 25.0 m. Substituting into the formula gives:
让我们用真实数值应用该模型：h₁ = 1.20 m，C = 0.12 m，c = 0.85 m，D = 25.0 m。代入公式得：
R = 0.12 − (1.20 + 0.12) × 0.85 / 25.0 ≈ 0.0751 m (about 7.5 cm)
om the first to the last row is therefore 2.18 m, which is feasible for a single tier of seating.
对于一个有30排的看台，最后一排的眼睛高度为h₃₀ = 1.20 + 29 × 0.0751 ≈ 3.38 m。从第一排到最后一排的垂直升高量为2.18 m，这对于单层看台是可行的。
6. Verifying Sightline Clearance for All Rows | 验证所有排的视线间隙
Although we designed R using the first row, it is important to check that the actual clearance Cₙ for higher rows does not fall below the required minimum. For row n, the line of sight to F at the position of row n−1 yields a height that can be computed. Substituting the arithmetic height hₙ shows that Cₙ increases with n, so the design is safe.
尽管我们使用第一行计算出了R，验证较高排的实际间隙Cₙ是否低于要求的最小值仍然很重要。对于第n排，可以计算其在第n−1排位置看向F的视线高度。将等差数列高度hₙ代入后可知Cₙ随着n增大而增加，因此设计是安全的。
7. Impact of Focal Distance D | 焦点距离D的影响
The chosen focal distance D drastically affects the required rise. If the focus is the nearest touchline, D may be only a few metres, making R very small or even negative in the formula above, which would be impossible. In practice, designers select a focal point further back on the pitch, such as the far goal line, or use a compromise distance to balance sightlines for different events. A larger D reduces the necessary R, leading to a shallower rake and lower overall height.
所选的焦点距离D会显著影响所需的升高量。如果焦点是最近的边线，D可能只有几米，使得上述公式中的R非常小甚至为负，这无法实现。实践中，设计者会选择场地中更远的焦点，例如远端球门线，或者使用一个折衷距离来平衡不同活动的视线。较大的D会减小所需的R，使坡度更缓，总高度更低。
8. Considering a Variable C‑value | 考虑可变的C值
Stadium guidelines such as the UK's Green Guide recommend a C‑value of 120 mm for optimal viewing. However, for venues with steep upper tiers or temporary seating, a smaller C‑value might be accepted, whereas for premium seats a larger clearance may be desired. By varying C in the formula, students can investigate the sensitivity of the rise and the total elevation of the stand. A larger C obviously demands a steeper rake and may lead to structural and safety challenges.
诸如英国《绿色指南》等体育场规范建议为获得最佳视野采用120 mm的C值。然而，对于有陡峭上层看台或临时座位的场馆，稍小的C值也可被接受，而对于高级座位则可能要求更大的间隙。通过改变公式中的C值，学生可以探究升高量和看台总高度的敏感性。显然，更大的C要求更陡的坡度，并可能带来结构和安全方面的挑战。
9. Seating Capacity and Total Height | 座位容量与总高度
e to surrounding structures. The arithmetic model allows quick estimation of capacity: if the horizontal depth available for seating is L, then N ≈ L / c, and the total height becomes H ≈ h₁ + (L/c − 1)R. These relationships help balance spectator comfort, safety, and construction cost.
一旦确定了每排升高量R，一个有N排的看台总高度为H = h₁ + (N − 1)R − 地面高度修正。对于给定的场地占地面积，可容纳的排数受到最大允许坡度（通常由建筑规范规定）以及相对于周边结构的总高度的限制。等差数列模型可以快速估算容量：若可用座位水平深度为L，则N ≈ L / c，总高度为H ≈ h₁ + (L/c − 1)R。这些关系有助于在观众舒适度、安全性和建造成本之间取得平衡。
10. Building an Excel Investigation Model | 建立Excel探究模型
n input cell), and then compute the sightline height at the previous row and the actual clearance Cₙ. Conditional formatting can highlight any rows where Cₙ drops below the required minimum. This dynamic model makes it easy to explore "what‑if" scenarios and reinforces the understanding of sequences.
电子表格非常适合探究D、c和C的变化如何影响排高度和间隙。学生可以建立如下栏目：排序号n，水平位置xₙ = (n−1)c，眼睛高度hₙ = h₁ + (n−1)R（R连接到输入单元格），然后计算在前一排处的视线高度以及实际间隙Cₙ。条件格式可以突出显示任何Cₙ低于要求最小值的排。这一动态模型便于探索“如果…会怎样”的情景，并加深对数列的理解。
11. Limitations and
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