Test: Measurements of Area & Volume - 1 - Civil Engineering (CE) MCQ

Test: Measurements of Area & Volume - 1 - Civil Engineering (CE) MCQ

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10 Questions MCQ Test Geomatics Engineering (Surveying) - Test: Measurements of Area & Volume - 1

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Test: Measurements of Area & Volume - 1 - Question 1

Which of the following formulas explains the calculation of area of earthwork using its mean depth? Area = BD + Sd2 Where, B = Breadth of section D = Mean depth of section Sd = Areas of sides

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 1

Methods for measurement of earthwork:
1. Mid Section method  2. Trapezoidal Method  3. Prismoidal Method  4. Simpson's 3/8th rule
Mid Section Method:
In this method the quantity of earthwork is computed with the help of size of mid section.

Test: Measurements of Area & Volume - 1 - Question 2

The formula used to calculate the mean depth of earthwork by averaging the depths of two consecutive sections is called:

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 2

Mid-Section Area Method

• The mean depth is the average depth of two consecutive sections.
• The area of mid-sections is calculated by using mean depth.
• The volume of the earthwork is calculated by multiplying the mid-section area by the distance between the two sections.

The formula used for finding the amount of earthwork using the mid-sectional method is given by:
Q = A x L
Where,
Q = Volume of earthwork
A = Area of mid-section = BDm + SDm2
B = width of the road
Dm = Depth of mid-section =
S = Slope
Hence, Q = (BDm + SDm2)L

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Test: Measurements of Area & Volume - 1 - Question 3

The chainages (m) and corresponding perpendicular offsets(m) taken from a chain line to an irregular boundary were (0, 0), (30, 2.65), (60, 3.80), (90, 3.75), (120, 4.65), (150, 3.60), (180, 5.00) and (210, 5.80). The area (m2) between the chain line and the irregular boundary by Simpson's rule is

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 3

Simpson's Rule:
In this rule, the boundaries between the ends of ordinates are assumed to form an arc of a parabola. Hence, Simpson's rule is sometimes called the parabolic rule.
The rule may be stated as follows,
To the sum of the first and the last ordinate, four times the sum of even ordinates and twice the sum of the remaining odd ordinates are added. This total sum is multiplied by the common distance. One-third of this product is the required area.
The area is given by Simpson's rule:

where O1, O2, O3, .........On is the offset
Limitation:
This rule is applicable only when the number of divisions is even, i.e., the number of ordinates is odd.
Calculation

Simpson's Rule :- If this rule is to be applied, the number of ordinates must be odd. But here the number of ordinate is even (eight)
so, Simpson's rule is applied from 01 to 07 and the area between 07 and 08 is found out by trapezoidal rule.

A1 = 10 × (5 + 40 + 16.9)
A1 = 619 m2
A= 30/2(5.00+5.80)
A2 = 162 m2
∴ Total area = A1 + A2
= 619 + 162 = 781 m2

Test: Measurements of Area & Volume - 1 - Question 4

For calculation of volume of earthwork, which of the following formulas assumes that short lengths in parabolic arcs are considered as parallel to each other?

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 4

Simpson's method assumes that the short lengths of the boundaries between the ordinates are parabolic arcs. This method is more useful when the boundary line departs considerably from the straight line.
Earthwork computation is involved in the excavation of channels, digging of trenches for laying underground pipelines, formation of bunds, and earthen embankments, digging of farm ponds, land leveling, and smoothening. In most of the computation, the cross-sectional areas at different interval along the length of the channels and embankments are first calculated and the volume of the prismoids are obtained between successive cross-section either by trapezoidal or prismoidal formula.
Calculation of area is carried out by any one of the following methods:
(a) Mid-ordinate method
(b) Average ordinate method
(c) Trapezoidal rule
(d) Simpson’s rule
Trapezoidal Rule

• Boundaries of the ordinates are presumed to be straight. Therefore the area between the lines is considered to be a trapezoid.

This rule can be used for any number of ordinates, and hence it has no drawbacks or limitations.
Simpson’s Rule:

• An arc is assumed to be present between the boundaries of the ordinates. Therefore it is also known as the parabolic rule.

• The only limitation to this rule is that it can only be used when the number of ordinates is odd.
• Simpson’s rule gives a more accurate result.
• After the calculation of the cross-sectional areas, volume is also needed to be calculated to determine the capacity
Test: Measurements of Area & Volume - 1 - Question 5

Prismoidal correction, while surveying is always?

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 5

The volume of earthwork by trapezoidal method  = V1

The volume of earthwork by prismoidal formula = V2

Prismoidal correction

• The volume by the prismoidal formula is more accurate than any other method
• But the trapezoidal method is more often used for calculating the volume of earthwork in the field.
• The difference between the volume computed by the trapezoidal formula and the prismoidal formula is known as a prismoidal correction.
• Since the trapezoidal formula always overestimates the volume, the prismoidal correction is always subtractive in nature is usually more than calculated by the prismoidal formula, therefore the prismoidal correction is generally subtractive.
• Volume by prismoidal formula = volume by the trapezoidal formula - prismoidal correction

Prismoidal correction (CP)

Where, D = Distance between the sections, S (Horizontal) : 1 (Vertical) = Side slope, d and d1 are the depth of earthwork at the centerline

Test: Measurements of Area & Volume - 1 - Question 6

The sum of first and last ordinates, add twice the sum of the remaining odd ordinates and four times the sum of all the even ordinates. The total sum thus obtained is multiplied by one-third of the common distance between the ordinates and the result gives the required area. This rule of finding the area is called:

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 6

Simpson's rule:
This rule is based on the assumption that the figures are trapezoids.
In order to apply Simpson's rule, the area must be divided in even number i.e., the number of offsets must be odd i.e., n term in the last offset 'On' should be odd.
The area is given by Simpson's rule:

Important Points

• In case of an even number of cross-sections, the end strip is treated separately and the area of the remaining strip is calculated by Simpson's rule. The area of the last strip can be calculated by either trapezoidal or Simpson's rule.
Test: Measurements of Area & Volume - 1 - Question 7

The Simpson’s rule for determination of areas is used when the number of offsets are:

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 7

Simpson's rule:
This rule is based on the assumption that the figures are trapezoids.
In order to apply Simpson's rule, the area must be divided in even number i.e., the number of offsets must be odd i.e., n term in the last offset 'On' should be odd.
The area is given by Simpson's rule:

where O1, O2, O3, .........On is the offset
Important Points

• In case of an even number of cross-sections, the end strip is treated separately and the area of the remaining strip is calculated by Simpson's rule. The area of the last strip can be calculated by either trapezoidal or Simpson's rule.
Test: Measurements of Area & Volume - 1 - Question 8

What is the volume of earthwork for constructing a tank that is excavated in the level ground to a depth of 4 m ? The top of the tank is rectangular in shape having an area of 50 m × 40 m and the side slope of the tank is 2: 1 (horizontal: vertical).

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 8
• Trapezoidal Formula:
Volume (v) of earthwork between a number of sections having areas A1, A2,…, An spaced at a constant distance d.

Also can be written as

Calculation:
Given, A1 = 50 × 40 = 2000 m2, As the side slope is given 2:1 i.e. H:V
So for a depth of 1 m, there is a change of 2 m in a Horizontal Direction.
So at 4 m vertical depth
Bottom Dimension is ( 50 - 16 ) = 34 m & (40 - 16) = 24 m
∴ The bottom area is 34 m × 24 m = 816 m2
Mean area (Am) = (2000 + 816) / 2 = 1408 m2

According to simple Trapezoidal rule for volume,

Test: Measurements of Area & Volume - 1 - Question 9

A road embankment 10 m wide at the formation level with side slopes 2:1 and with an average height of 5 m is constructed with an average gradient of 1:40 from the contour 220 m to 280 m. Find the volume of earthwork.

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 9

A gradient is the rate of rise or falls along the length of the road with respect to horizontal. It is expressed as ‘1' vertical unit to 'N' horizontal units.

Area of trapezoidal:
According to the trapezoid area formula, the area of a trapezoid is equal to half the product of the height and the sum of the two bases.
Area = ½ x (Sum of parallel sides) x (perpendicular distance between the parallel sides).
Calculation:

Average height = 5 m
Difference in elevation(h) = 280 - 220 = 60 m

Average cross-sectional area(A) = (1/2) ×(10+30)×5
A = 100 m2
Volume of earthwork = A × L
Volume of earthwork = 100 × 2400
∴ Volume of earthwork = 2,40,000 m3

Test: Measurements of Area & Volume - 1 - Question 10

The areas enclosed by the contours in a lake are as follows:

The volume of water between the contours 270 m and 290 m by trapezoidal formula is _______.

Detailed Solution for Test: Measurements of Area & Volume - 1 - Question 10

Trapezoidal Formula:
Volume (v) of earthwork between a number of sections having areas A1, A2, …, An spaced at a constant distance d.

Simpson’s Formula:
Volume (v) of the earthwork between a number of sections having area A1, A2, …, An spaced at constant distance d apart is

Calculation:
The areas enclosed by the contours in a lake are as follows:

Given contour interval (d) = 275-270 = 5 m
So using trapezoidal formula:
8000 m3

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