Problem 17 - Basic Technical Mathematics with Calculus

Question

Calculate the indicated areas. All data are accurate to at least two significant digits. Soundings taken across a river channel give the following depths with the corresponding distances from one shore. $$\begin{array}{l|l|l|l|l|l|l|l|l|l|l|l}\text {Distance (ft)} & 0 & 50 & 100 & 150 & 200 & 250 & 300 & 350 & 400 & 450 & 500 \\\\\hline \text {Depth (ft)} & 5 & 12 & 17 & 21 & 22 & 25 & 26 & 16 & 10 & 8 & 0\end{array}$$ Find the area of the cross section of the channel using Simpson's rule.

Step-by-Step Solution

Answer

The area of the cross-section of the channel is 8050 square feet.

Step 1: Understanding Simpson's Rule

Simpson's Rule is a method for approximating the integral of a function, typically used to find the area under a curve. The formula for Simpson's Rule when applied to equally spaced data points is given by: \[ A = \frac{h}{3} \left[ y_0 + 4(y_1 + y_3 + y_5 + ...) + 2(y_2 + y_4 + y_6 + ...) + y_n \right] \] where \( h \) is the interval between each \( x \)-value (in this case, the distance), \( y \) values are the depths, and \( n \) is the total number of intervals, which should be an even number.

Step 2: Determine the Intervals and Assign Values

Given the data, there are 11 depths (12 total distances). This divides the distances into 10 intervals. We can check that the intervals between the distances are consistent: \( h = 50 \) feet, and since there are 11 distances, there will be 10 intervals, which is even and suitable for Simpson's Rule.

Step 3: Apply Simpson's Rule Formula

Plug the depths into Simpson's Rule formula. Compute each part separately. \(A = \frac{50}{3} \left[ 5 + 4(12 + 21 + 25 + 16 + 8) + 2(17 + 22 + 26 + 10) + 0 \right] \) \(= \frac{50}{3} \left[ 5 + 4(82) + 2(75) + 0 \right] \)

Step 4: Simplify Each Term

First, compute the sums: \(4(82) = 328\) and \(2(75) = 150\).Substitute these into the equation:\[ A = \frac{50}{3} [5 + 328 + 150 + 0] \] Simplify further: \(A = \frac{50}{3} [483]\).

Step 5: Final Calculation

Calculate the numerical value: \( \frac{50}{3} \times 483 = 8050 \) This gives \( A = 8050 \) square feet, which is the area of the cross-section of the channel.

Key Concepts

Numerical Integration

Numerical integration is a core technique in calculus used to compute an approximation of the integral. It is particularly useful when dealing with real-world data where functions are not easily integrated using traditional techniques. Simpson's Rule is one of the most popular numerical integration methods.

• Simpson's Rule offers a polynomial approximation, which can prove more accurate than other methods like the Trapezoidal Rule.
• It is especially used for evenly spaced data points, which is often how data from physical measurements is collected, such as in the context of measuring river depths across predefined distances.

Understanding how to apply numerical integration methods like Simpson's Rule can simplify dealing with complex functions and irregularly shaped areas, transforming difficult calculus problems into more digestible arithmetic ones.

Area Under a Curve

Calculating the area under a curve is essential in many fields, including math, physics, and engineering. When you plot a graph of function, the area under the curve between two points on the x-axis represents a definite integral, and it has real-world applications like finding out the space between riverbanks.

To determine the area under a curve using numerical integration, specifically Simpson's Rule, you follow these steps:

• Identify evenly spaced intervals across the x-axis. Here, the distance between each river measurement point is 50 ft.
• Plug these data points into the Simpson's Rule formula, which considers the heights (depths) at each interval.
• Sum up areas of the segments to find the total area under the curve.

This method of calculating the area becomes highly useful when the function forms a curve that does not attach easily to standard geometrical shapes.

Calculus

Calculus, the mathematics of change, is fundamental to understanding motion, growth, and other dynamic processes. In this exercise, calculus provides the framework to understand and compute changes in the depth of a river using real data.

A few fundamental concepts of calculus related to this exercise include:

• Definite Integrals: Used to compute total quantities, such as cross-sectional areas, over a specific interval.
• Approximation Techniques: Methods like Simpson's Rule allow us to calculate integrals using polynomial approximations when exact answers are difficult.

Applying calculus in practical scenarios, such as finding the cross-sectional area of a river channel, helps in transforming theoretical data into actionable insights. It bridges the gap between abstract mathematics and tangible, real-world applications.

Cross-sectional Area Calculation

The cross-sectional area calculation is important in various engineering and environmental studies, as it helps measure the flow capacity of natural and artificially constructed channels such as rivers, ducts, and piping systems.

For a practical cross-sectional area calculation using Simpson's Rule:

• Data Gathering: Start by collecting depth measurements across a channel at regular intervals.
• Apply the Formula: Use the Simpson's Rule formula, substituting intervals ( 50 ft in this case) and the depths to find the area under the curve that represents the cross-section.

This calculation helps in understanding the river's capacity to transport water, critical for flood management and designing structures such as bridges and canals.