Question

In case of prismoidal rule of determining the capacity of pond followings are used
A. The number of contours should be odd
B. The number of contour intervals should be odd
C. The number of contours should be even
D. The number of contour intervals should be even
Choose the correct answer from the options given below:

• A. A and B only
• B. A and D only
• C. B and C only
• D. C and D only

Hint:

Think of the simplest case: a single prismoid has 3 cross-sections (top, middle, bottom - an odd number of areas) and 2 intervals (even number of intervals). This makes it easy to remember the rule.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The prismoidal rule (derived from Simpson's 1/3rd rule) is used to estimate the volume of earthwork, reservoirs, or ponds between parallel contour planes.
For the mathematical formulation of this rule to be valid, specific conditions regarding the number of boundary planes and intervals must be met.

Step 2: Detailed Explanation:
The formula for the prismoidal rule for volume calculation between parallel areas is:
\[ V = \frac{d}{3} [ (A_1 + A_n) + 4(A_2 + A_4 + \dots + A_{n-1}) + 2(A_3 + A_5 + \dots + A_{n-2}) ] \]
Where:
- \(d\) is the constant contour interval.
- \(A_1, A_2, \dots, A_n\) are the cross-sectional or contour areas.
For this mathematical division into triplets to work, the total number of contour areas (\(n\)) must be an odd number (A).
When the total number of contours is odd, the number of spaces or intervals between them (which is equal to \(n-1\)) is an even number (D).
If there is an even number of contours, the prismoidal rule cannot be applied to the entire series directly; the last interval must be computed separately using the trapezoidal rule.
Therefore, the prismoidal rule requires an odd number of contours and an even number of contour intervals.

Step 3: Final Answer:
The correct combination is A and D only.