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Interactive Audio Lesson
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Area Calculations
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Today, we are going to explore area calculations in surveying, particularly using the Trapezoidal and Simpson’s rules. Can anyone tell me the basic principle behind these methods?
The Trapezoidal rule estimates the area by dividing it into trapezoids, while Simpson's rule uses parabolic segments to get a more accurate area calculation.
Exactly! The Trapezoidal rule is simpler and helps when we have straight sections, while Simpson's is used for smoother curves. For example, in Example 1.19, we used these methods to find areas between a chain line and an irregular boundary. Let's summarize the calculations.
In the Trapezoidal method, we used the formula A = (d/2)(y1 + y2) where d is the distance between points. Is that correct?
Yes! And for Simpson's rule, we utilized A = (d/3)(y0 + 4y1 + 2y2 + ... + yn). It's essential to remember those formulas. Can anyone provide a quick mnemonic for remembering them?
Maybe something like 'Trap Averages' for Trapezoidal and 'Simpson's Super Summation' for Simpson's rule?
Great suggestions! To summarize, both methods provide intuitive ways of estimating areas, with Simpson's being more accurate for real-world applications involving curves.
Calculating Volume of Earthworks
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Next, let's dive into computing earthwork volumes. In Example 1.20, we considered an embankment and calculated the volume using trapezoidal and prismoidal formulas. Who remembers the difference between these two approaches?
The trapezoidal formula takes the average of cross-sections, while the prismoidal formula adjusts for variations between the sections.
Exactly! When we apply these formulas, we not only ensure accurate estimations but also provide essential data for project planning. Let's look at the calculations in detail. How do we find cross-sectional areas?
We use the formula Δ = (b + sh) * h, where b is the base width, sh is the side slope height, and h is the height.
Right! It's crucial to accurately calculate those cross-sections before plugging them into the volume formulas. To recap, understanding both approaches helps in larger-scale civil projects.
Tacheometry and Elevation Determination
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Finally, let's discuss tacheometry, as seen in Examples 1.22 to 1.30. This technique allows us to measure distances indirectly using readings from a tacheometer. Can anyone describe how these measurements work?
The measurements depend on staff readings and the angles of elevation or depression to a staff held vertical.
Correct! For instance, in Example 1.22, we calculated horizontal distances using the formula D = KS + C. Can anyone summarize how K and C are determined?
K is the constant that helps convert staff readings into distances, and C is the additive constant obtained from the initial measurements.
Well said! Keep in mind that accurately determining these constants is essential for precise surveying. We frequently use these same principles in practice, especially for larger surveying projects.
Introduction & Overview
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Quick Overview
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The section explores several practical examples demonstrating the application of surveying formulas, including the Trapezoidal and Simpson’s rules for calculating area, as well as methods of computing earthwork volumes and determining elevations through tacheometry.
Detailed
Detailed Summary
This section focuses on practical examples and solutions concerning various surveying techniques used to calculate areas, volumes, and elevations. The examples include:
- Area Calculations: Examples 1.19 to 1.21 illustrate how to compute areas between irregular boundaries and contours using both the Trapezoidal and Simpson’s rules, demonstrating the application of mathematical principles in real-world scenarios.
- Earthwork Volume Calculation: Example 1.20 discusses the estimation of the volume of earthwork required for an embankment using the trapezoidal and prismoidal formulas, emphasizing the importance of cross-sectional area calculations.
- Tacheometric Measurements: Examples 1.22 to 1.30 delve into using a tacheometer for measuring distances and elevations, illustrating how horizontal distances can be determined based on staff readings and angles of depression or elevation.
These examples are pivotal in understanding the various techniques in surveying and their applications in engineering projects.
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Example 1.19: Area Calculation Using Different Rules
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The following offsets were taken from a chain line to an irregular boundary line at an interval of 10 m: 0, 2.50, 3.50, 5.00, 4.60, 3.20, and 0 m. Compute the area between the chain line, the irregular boundary line and the end of offsets by: (a) the Trapezoidal rule, and (b) Simpson’s rule.
Solution:
By Trapezoidal rule: Required area=10 / 2{0 + 0 + 2(2.50 + 3.50 + 5.00 + 4.60 + 3.20)} = 537.60 =188 m²
By Simpson’s rule: Required area=10/ 3{0 + 0 + 4(2.50 + 5.00 + 3.20) + 2(3.50 + 4.60)} = 10/3{42.80 +16.20}=10/359.00 = 196.66 m²
Detailed Explanation
In Example 1.19, we learn to compute areas using two different numerical methods: the Trapezoidal rule and Simpson's rule. The trapezoidal rule calculates area by approximating the shape under the curve with trapezoids, while Simpson's rule uses parabolic segments for better accuracy.
- Trapezoidal Rule: We sum the areas of trapezoids formed between the chain line and the irregular boundary. The formula is simplified by summing offset values.
- Simpson’s Rule: This rule is more accurate as it fits parabolas to the data points. It requires both odd and even indexed offsets and uses a similar approach of summation but with different weights.
The final calculation results show how different methods can reach slightly varied results, providing insight into numerical analysis for area calculations.
Examples & Analogies
Imagine a farmer wanting to find the area of an irregular field. Using the trapezoidal method is like putting down small rectangular plots to cover the entire field and calculating their areas. Simpson's method is like using curved, flexible tape to outline the grass, which allows for a deeper and more accurate assessment of the area because it adheres to the edges better.
Example 1.20: Earthwork Volume Calculation
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An embankment of width 10 m and side slopes 1½:1 is required to be made on a ground which is level in a direction transverse to the centre line. The central heights at 40 m intervals are as: 0.90, 1.25, 2.15, 2.50, 1.85, 1.35, and 0.85 m, calculate the volume of earth work according to (i) Trapezoidal formula, and (ii) Prismoidal formula.
Solution:
The cross-sections areas are calculated by ∆= (b + sh)h
- ∆1= (10 + 1.50.90)0.90 = 10.22 m²
- ∆2= (10 + 1.51.25)0.90 = 14.84 m²
- ∆3= (10 +1.51.25)2.15 = 28.43 m²
- ∆4= (10 +1.52.50)2.50 = 34.38 m²
- ∆5= (10 +1.51.85)1.85 = 23.63 m²
- ∆6= (10 +1.51.35)1.35 = 16.23 m²
- ∆7= (10 +1.50.85)*0.85= 9.58 m²
(a) Volume according to trapezoidal formula V= 40 / 2{10.22 + 9.58 + 2(14.84 + 28.43 + 34.38 + 23.63 + 16.23)} = 5096.4 m³
(b) Volume calculated in prismoidal formula: V = 40 / 3 {10.22 + 9.58 + 4(14.84 + 34.38 +16.23) + 2(28.43 + 23.63)} = 5142.9 m³
Detailed Explanation
In Example 1.20, we calculate the volume of earthwork for constructing an embankment using two different methods: Trapezoidal and Prismoidal formulas.
- Cross-section Area Calculation: First, we calculate the cross-sectional areas based on the widths and heights at specified intervals, considering the side slopes of the embankment.
- Trapezoidal Volume Calculation: The trapezoidal method merely uses the average of these cross-sectional areas multiplied by the length.
- Prismoidal Volume Calculation: This more accurate method gives additional weight to the more centrally located cross-section, enhancing the volume calculation precision.'
Examples & Analogies
Think of filling a sandbox with sand where the heights of sand differ at different sections. The trapezoidal method is like taking the average height of different sections to estimate how much sand you'll need, while the prismoidal method adjusts for the central height more accurately, ensuring none of the unevenness is overlooked.
Example 1.21: Volume Calculation Between Contours
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The areas enclosed by the contours in the lake are as follows:
Contour (m) Area (m²)
270 2050
275 8400
280 16300
285 24600
290 31500
Calculate the volume of water between the contours 270 m and 290 m by Trapezoidal formula.
Solution: Volume according to trapezoidal formula: = 5 / 2{2050 + 31500 + 2(8400 + 16300 + 24600)} = 330,250 m³
Detailed Explanation
In Example 1.21, we calculate the volume of water between two contour levels using the trapezoidal formula. Each contour represents a different elevation level, and we know their respective enclosed areas.
- Areas of Contours: Here, we take areas corresponding to different height levels of the lake and recognize that the volume beneath these contour levels is not uniform.
- Volume Calculation: Applying the trapezoidal formula allows us to approximate the volume of water between the specified contours effectively, providing critical information for maintaining water levels or planning drainage.
Examples & Analogies
Imagine a multi-layered cake where each layer represents a contour level of a lake. While calculating the volume of cake, we would combine the areas of each layer, just like using the trapezoidal formula to estimate the water volume between the layers accurately.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Area Calculations: Techniques like Trapezoidal and Simpson's Rule are crucial for finding areas in surveying.
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Earthwork Volume: Understanding how to compute volumes of earthwork is essential for construction projects.
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Tacheometry: This method helps achieve accurate distance measurements and is vital in various surveying applications.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example 1.19 demonstrates area calculation using both Trapezoidal and Simpson’s rules.
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In Example 1.20, earthwork volumes for an embankment are calculated using the trapezoidal and prismoidal methods.
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Example 1.22 showcases the application of tacheometry to determine distances and elevations.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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To find the area, don't be shy, divide by two and give it a try!
📖 Fascinating Stories
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Imagine a calm lake where the sun sets, you can measure the shoreline with gentle steps, using trapezoids to find your way, counting areas under the sky's broad play.
🧠 Other Memory Gems
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A.T.E. - Area Trapezoidal Estimation, for remembering area calculations with trapezoids.
🎯 Super Acronyms
T.S.R. - Trapezoidal, Simpson, Rule, to remember the two primary area calculation techniques.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Trapezoidal Rule
Definition:
A numerical method for estimating the area under a curve by dividing it into trapezoids.
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Term: Simpson’s Rule
Definition:
A method for calculating the area under a curve using parabolic segments, providing better accuracy than the trapezoidal rule.
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Term: Tacheometry
Definition:
A method of distance measurement in surveying that determines the distance from the instrument to a point based on the staff readings.
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Term: Earthwork Volume
Definition:
The volume of soil that is excavated or filled during construction processes.
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Term: Crosssectional Area
Definition:
The area of a section taken through a solid object, which is used in calculating volumes in earthworks.