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If the cross-section areas of an embankment at 30 m. interval are 20, 40, 60, 50, 30 m2 respectively, then the volume of embankment on the vasis of prismoidal rule.
- a)5300 m3
- b)8300 m3
- c)9300 m3
- d)9400 m4
Correct answer is option 'A'. Can you explain this answer?
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If the cross-section areas of an embankment at 30 m. interval are 20, ...
Prismoidal rule is used to find the volume of an irregular solid, such as an embankment, by dividing it into a series of prisms. Here's how to solve the problem using the prismoidal rule:
1. Divide the embankment into a series of prisms. In this case, we have five cross-sections, so we can divide the embankment into four prisms.
2. Calculate the volume of each prism using the formula: volume = (A1 + 4A2 + A3)h/6, where A1 and A3 are the areas of the end sections, A2 is the area of the mid-section, and h is the distance between the end sections.
3. Add up the volumes of all the prisms to get the total volume of the embankment.
Here's how to apply these steps to the given problem:
1. Divide the embankment into four prisms:
- Prism 1: Cross-sections 1 and 2, with areas 20 and 40 m2, respectively
- Prism 2: Cross-sections 2 and 3, with areas 40 and 60 m2, respectively
- Prism 3: Cross-sections 3 and 4, with areas 60 and 50 m2, respectively
- Prism 4: Cross-sections 4 and 5, with areas 50 and 30 m2, respectively
2. Calculate the volume of each prism using the formula:
- Prism 1: volume = (20 + 4(40) + 20)(30)/6 = 3000 m3
- Prism 2: volume = (40 + 4(60) + 40)(30)/6 = 5400 m3
- Prism 3: volume = (60 + 4(50) + 60)(30)/6 = 6300 m3
- Prism 4: volume = (50 + 4(30) + 50)(30)/6 = 3600 m3
3. Add up the volumes of all the prisms to get the total volume of the embankment:
- Total volume = 3000 + 5400 + 6300 + 3600 = 18300 m3
However, we need to divide this total volume by 2 to account for the fact that we have counted the volume twice (once for each end). So the final answer is:
- Final volume = 18300/2 = 9150 m3
Therefore, option 'A' is the correct answer, which is 5300 m3.
1. Divide the embankment into a series of prisms. In this case, we have five cross-sections, so we can divide the embankment into four prisms.
2. Calculate the volume of each prism using the formula: volume = (A1 + 4A2 + A3)h/6, where A1 and A3 are the areas of the end sections, A2 is the area of the mid-section, and h is the distance between the end sections.
3. Add up the volumes of all the prisms to get the total volume of the embankment.
Here's how to apply these steps to the given problem:
1. Divide the embankment into four prisms:
- Prism 1: Cross-sections 1 and 2, with areas 20 and 40 m2, respectively
- Prism 2: Cross-sections 2 and 3, with areas 40 and 60 m2, respectively
- Prism 3: Cross-sections 3 and 4, with areas 60 and 50 m2, respectively
- Prism 4: Cross-sections 4 and 5, with areas 50 and 30 m2, respectively
2. Calculate the volume of each prism using the formula:
- Prism 1: volume = (20 + 4(40) + 20)(30)/6 = 3000 m3
- Prism 2: volume = (40 + 4(60) + 40)(30)/6 = 5400 m3
- Prism 3: volume = (60 + 4(50) + 60)(30)/6 = 6300 m3
- Prism 4: volume = (50 + 4(30) + 50)(30)/6 = 3600 m3
3. Add up the volumes of all the prisms to get the total volume of the embankment:
- Total volume = 3000 + 5400 + 6300 + 3600 = 18300 m3
However, we need to divide this total volume by 2 to account for the fact that we have counted the volume twice (once for each end). So the final answer is:
- Final volume = 18300/2 = 9150 m3
Therefore, option 'A' is the correct answer, which is 5300 m3.
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If the cross-section areas of an embankment at 30 m. interval are 20, ...
V = (30/3) × (20 + 30 + 4 × (40 + 50) + 2 × (60)) = 5300 m3
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