2.2. Long Questions

Example 1.19: 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.

Solution: By Trapezoidal rule: Here d=10 m Required area=10 / 2{0 + 0 + 2(2.50 + 3.50 + 5.00 + 4.60 + 3.20)} = 5*37.60 =188 m2

In this example, we calculate the area under a curve defined by a series of offsets (measurements) taken at regular intervals.

1. Dataset Interpretation: The measurements are offsets from a chain line to a boundary. To find the area, we use the Trapezoidal Rule, which approximates the area under the curve with trapezoids.
2. Setting Up the Formula: The formula is based on the sum of the first and last heights and twice the sum of the intermediate heights multiplied by the width of each trapezoid (d).
3. Substituting Values: Here, d is 10 meters, and we have two heights (0 and 0) plus the offsets from the chain as 2.50, 3.50, etc. We solve this by plugging in the values.
4. Calculating: The area is calculated as 188 m², which tells us the total area encompassed by the curve.

Imagine you're in a park and have a jogging path that bends in and out—analogous to our series of offsets. If you want to know how much ground you cover, you could visualize each segment of your path (where you stop to turn) as a trapezoid. The further apart you measure, the rougher your estimate would be—this is similar to how the Trapezoidal Rule works.

(b) Simpson’s rule.

Solution: By Simpson’s rule: d=10 m Required area=10/ 3{0 + 0 + 4(2.50 + 5.00 + 3.20) + 2(3.50 + 4.60)} = 10/3{42.80 +16.20}=10/3*59.00 = 196.66 m2

Similar to the Trapezoidal Rule, Simpson's Rule is a method used to calculate the area under a curve, but it often provides a more accurate estimate by fitting parabolas instead of straight lines between points.

1. Understanding Simpson's Rule: The area is calculated considering weights to each point based on its position. Points that are closer to the middle (like the middle two points) are given more weight (4x) compared to end points (1x).
2. Identify Measurements: Here, the offsets are still the same, taken at consistent intervals.
3. Applying the Formula: We apply the measurement values into Simpson's formula and calculate.
4. Final Area Calculation: The calculated area using Simpson's Rule gives us a more refined total area of 196.66 m² compared to the trapezoidal method.

Think of this process as trying to calculate the area of an irregular field. Instead of just measuring straight lines (which misses dips and rises), you're trying to take a more detailed approach by sampling more frequently. In a way, you're creating gentle curves through the data points, akin to driving around a winding road rather than taking a straight shortcut.

Example 1.20: An embankment of width 10 m and side slopes 1½:1 is required to be made on 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.

In this example, we are tasked with calculating the volume of earth that needs to be moved to create an embankment. The key points are:

1. Understanding Measurements: The width of the embankment is uniform (10m), and the heights at intervals give the necessary depth of the material needed.
2. Calculating Cross-Section Areas: As we move along the embankment, we calculate the area of each cross-section using the formula for the area of a trapezoid or triangle (taking into account the side slopes).
3. Applying Formulas: Both the Trapezoidal and Prismoidal formulas are used to compute total volume using the area of those cross-sectional areas along the length (40 m in this case).
4. Volume Results: The total volume calculated will differ depending on the formula—this reflects how each method approximates the actual volume of earth.

If you've ever dug a trench in your garden to install a small fence, you have experienced a similar calculation. You can think of each segment of the trench as a cross-section of varying width, making sure you understand how deep to dig based on the existing soil height.

Example 1.21: 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

In this scenario, we’re trying to find out how much water could fit between certain contour lines around a lake. Each contour represents the water level, so the areas associated with each contour help us establish the volume:

1. Defining Contours: Each number represents a different water level, and the areas show how much of the lake’s surface is at that level.
2. Using the Formula: The Trapezoidal formula is applied similarly to previous examples, treating the contour areas as trapezoidal slices of the lake.
3. Calculation Method: Calculate using the areas for the contour heights to get the total volume of water contained between the specified contour levels.
4. Resulting Volume: The final result is 330,250 m³, giving us the capacity of the lake section between those two heights.

Imagine layering liquid in a series of bowls where each contour level is the top of a bowl. If you pour different amounts of water (regions) into the bowls, the total amount of water can be calculated by measuring how much is in between the bowls—noticing that the more areas you account—for each level, the more accurate your total will be.

--