题目
A river is being studied.
At one particular place, the river is 15 m wide.
The depth, metres, of the river is measured at a point metres from one side of the river.
Figure 1 shows a plot of the cross-section of the river and the coordinate values .
(a) Use the trapezium rule with all the values given in Figure 1 to estimate the cross-sectional area of the river.
The water in the river is modelled as flowing at a constant speed of m s across the whole of the cross-section.
(b) Use the model and the answer to part (a) to estimate the volume of water flowing through this section of the river each minute, giving your answer in m to 2 significant figures.
Assuming the model,
(c) state, giving a reason for your answer, whether your answer for part (b) is an overestimate or an underestimate of the true volume of water flowing through this section of the river each minute.
题目中文翻译
一条河流正在被研究。
在某一特定位置，河流宽15米。
河流的深度 （米）是在距河流一侧 （米）处测量的。
图1显示了河流横截面的图形以及坐标值 。
(a) 使用图1中给出的所有 值，用梯形法则估算河流的横截面积。
河水以恒定速度 m/s 流过整个横截面。
(b) 利用模型和(a)的答案，估算每分钟流过该河段的水的体积，答案以 m 为单位，保留2位有效数字。
(c) 指出(b)的答案是高估还是低估了真实值，并说明理由。
（6分）
解答
解法一
思路
利用梯形法则估算河流横截面积。梯形公式为 。
答题过程
(a) From Figure 1, the width of the river is m and the values are measured at equally spaced intervals. Therefore .
Applying the trapezium rule:
A &\approx \frac{3}{2}\left[0 + 2(1.52 + 2.74 + 3.12 + 3.08) + 0\right] \\[2mm] =&\, \frac{3}{2} \times 2 \times 10.46 \\ =&\, 31.38 \text{ m}^2 \end{align*}$$ $$\boxed{A = 31.38 \text{ m}^2}$$ **(b)** The water flows at a constant speed of $1.5$ m s$^{-1}$. The volume per minute is: $$\begin{align*} V =&\, 31.38 \times 1.5 \times 60 \\ =&\, 2824.2 \\ &\approx 2800 \text{ m}^3 \quad (2 \text{ s.f.}) \end{align*}$$ $$\boxed{V \approx 2800 \text{ m}^3}$$ **(c)** This is an **underestimate**. The trapezium rule approximates the curve by straight line segments. Since the river bed is concave (curves below the straight lines), the trapezia have smaller area than the actual cross-section.