Problem 2 - Basic Engineering Mathematics

Question

A river is \(15 \mathrm{~m}\) wide. Soundings of the depth are made at equal intervals of \(3 \mathrm{~m}\) across the river and are as shown below. \(\begin{array}{llllllll}\text { Depth }(\mathrm{m}) & 0 & 2.2 & 3.3 & 4.5 & 4.2 & 2.4 & 0\end{array}\) Calculate the cross-sectional area of the flow of water at this point using Simpson's rule.

Step-by-Step Solution

Answer

The cross-sectional area is 51.4 square meters.

Step 1: Introduction to Simpson's Rule

Simpson's Rule is a method for estimating the value of an integral. It is particularly useful when data points are spaced at even intervals. Given that the river width is divided into equal segments of 3 meters and the depths at these intervals are provided, we can use Simpson's rule to calculate the cross-sectional area.

Step 2: List Given Data

Here are the depth measurements at intervals of 3 meters across the river: \( y_0 = 0, y_1 = 2.2, y_2 = 3.3, y_3 = 4.5, y_4 = 4.2, y_5 = 2.4, y_6 = 0 \). The width \( h \) between each interval is 3 meters.

Step 3: Apply Simpson's Rule Formula

Simpson's Rule formula for approximate integration is: \[\frac{h}{3} (y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + 4y_5 + y_6)\]. Plug in the given values for the depth.

Step 4: Calculate Using Formula

Substitute the values: \( h = 3 \), \( y_0 = 0 \), \( y_1 = 2.2 \), \( y_2 = 3.3 \), \( y_3 = 4.5 \), \( y_4 = 4.2 \), \( y_5 = 2.4 \), and \( y_6 = 0 \). Calculate the expression: \[\frac{3}{3} (0 + 4 \times 2.2 + 2 \times 3.3 + 4 \times 4.5 + 2 \times 4.2 + 4 \times 2.4 + 0)\].

Step 5: Evaluate the Expression

First solve inside the parentheses:\(0 + 4 \times 2.2 = 8.8\), \(2 \times 3.3 = 6.6\), \(4 \times 4.5 = 18\), \(2 \times 4.2 = 8.4\), \(4 \times 2.4 = 9.6\).Now sum them up: \(8.8 + 6.6 + 18 + 8.4 + 9.6 = 51.4\).

Step 6: Compute Final Area

Replace back: \[1 \times 51.4\]. So the cross-sectional area is 51.4 square meters.

Key Concepts

Numerical Integration

Numerical integration is a core concept in mathematics that focuses on finding the approximate value of an integral when an analytical solution is challenging or impossible to obtain directly. This is especially useful when dealing with engineering and physical problems where measurements are taken at discrete intervals, just like in our exercise.

Simpson's Rule is one such method used to carry out numerical integration. It is specifically designed for evenly spaced data points. The rule approximates the area under the curve by using segments of parabolas rather than straight-line segments, providing a more accurate approximation compared to other methods like the trapezoidal rule.

To use Simpson's Rule, one must have data points that are equidistant from each other. This method represents a functional relationship by summing up contributions from sampled data using weighted coefficients. By doing so, it achieves higher accuracy, particularly for smooth data representations. The ability to effectively approximate integrals makes it indispensable in engineering, science, and many other fields.

Cross-sectional Area

Calculating the cross-sectional area is essential in various engineering disciplines, especially in fluid dynamics and structural analysis. A cross-sectional area helps in understanding the distribution of a given quantity, such as fluid flow through a channel or air through a duct.

In our exercise, the cross-sectional area of a river is calculated using depth measurements taken at regular intervals. This involves taking measurements across the width of the river and applying a numerical method to estimate the area, which in this case is Simpson's Rule. For our scenario, the cross-sectional area relates directly to how much water can flow through that section of the river, making it crucial for assessing the river's capacity and potential applications such as irrigation or flood prediction.

Accurate estimation of the cross-sectional area is vital because it determines constraints and capabilities in hydraulic engineering projects. The calculated area takes the shape of the actual depth profile, thereby providing valuable insights into potential river behavior and engineering solutions.

Engineering Mathematics

Engineering mathematics involves the application of mathematical techniques and concepts to solve complex engineering problems. It serves as the backbone for designing, analyzing, and improving engineering processes and structures.

In scenarios like our exercise, where systematic calculation of fluid flow and river depth is needed, engineering mathematics provides the necessary tools to derive meaningful conclusions from basic data. Understanding and applying integration techniques, like Simpson's Rule, allows engineers to measure real-world phenomena that are critical for designing efficient systems.

More broadly, engineering mathematics includes various disciplines such as calculus, linear algebra, probability, and statistics. Each branch has its role in solving specific engineering problems—whether it's calculating structural loads, predicting system performances, or optimizing processes. By using approaches derived from numerical methods, engineers can create models to simulate real-world situations, paving the way for innovation and technological advancement.