Well Digging: Volume & Plastering Cost Calculation

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Alright, let's break down this math problem step by step, guys! We've got a well to dig, and we need to figure out how much earth we're going to remove and how much it will cost to plaster the inside. This involves calculating the volume of a cylinder (the well) and the curved surface area. Don't worry; we'll make it super clear and easy to follow.

Calculating the Volume of Earth Dug Out

First, we need to find out the volume of earth that needs to be dug out. Since the well is cylindrical, we'll use the formula for the volume of a cylinder, which is:

Volume = π * r² * h

Where:

  • Ï€ (pi) is approximately 3.14159
  • r is the radius of the well
  • h is the depth (height) of the well

We're given the diameter of the well, which is 7 meters. To find the radius, we simply divide the diameter by 2:

r = 7 m / 2 = 3.5 m

We're also given the depth of the well, which is 22.5 meters. Now we have all the values we need to calculate the volume:

Volume = π * (3.5 m)² * 22.5 m Volume = 3.14159 * (12.25 m²) * 22.5 m Volume = 3.14159 * 275.625 m³ Volume ≈ 865.90 m³

So, you'll need to dig out approximately 865.90 cubic meters of earth. Remember this number, as it tells us the sheer amount of digging we're talking about! Imagine all that soil being removed – quite a task, isn't it? This volume is crucial for planning the excavation and understanding the scale of the project. It's also essential for logistical considerations such as where to store the excavated soil or how many trucks are needed to haul it away. Now, let's move on to the next part of the problem: figuring out the cost of plastering the inner curved surface.

Why is the Volume Calculation Important?

Understanding the volume of earth that needs to be removed is essential for several reasons. First and foremost, it helps in estimating the labor and equipment required for the job. Knowing the volume allows you to determine the size of the excavation team and the type of machinery needed, such as excavators or backhoes. A precise volume calculation ensures that you have the necessary resources to complete the digging efficiently and without unexpected delays.

Moreover, the volume calculation is crucial for budgeting purposes. By knowing the amount of earth to be removed, you can accurately estimate the cost of disposal. Excavated soil needs to be stored or transported away from the site, and this process incurs expenses. The volume calculation helps in determining the number of truckloads required, the distance to the disposal site, and the associated transportation costs. Accurate budgeting prevents cost overruns and ensures that the project stays within its financial limits.

Furthermore, the volume calculation plays a vital role in environmental planning. Excavating large volumes of earth can have significant environmental impacts, such as soil erosion and habitat disruption. By knowing the volume of earth to be removed, you can develop strategies to mitigate these impacts. This may involve implementing erosion control measures, such as silt fences or vegetation cover, to prevent soil loss. It may also involve careful planning of the excavation process to minimize disturbance to surrounding habitats.

In addition to these practical considerations, the volume calculation is also important for regulatory compliance. Many jurisdictions have regulations governing excavation activities, including requirements for soil disposal and environmental protection. Accurate volume calculations are necessary to comply with these regulations and avoid potential fines or legal issues. By demonstrating a thorough understanding of the volume of earth to be removed, you can ensure that the project meets all applicable regulatory requirements.

Calculating the Cost of Plastering the Inner Curved Surface

Next, we need to calculate the cost of plastering the inner curved surface of the well. The formula for the curved surface area of a cylinder is:

Curved Surface Area = 2 * π * r * h

Where:

  • Ï€ (pi) is approximately 3.14159
  • r is the radius of the well
  • h is the depth (height) of the well

We already know the radius (3.5 m) and the depth (22.5 m), so we can plug these values into the formula:

Curved Surface Area = 2 * 3.14159 * 3.5 m * 22.5 m Curved Surface Area = 2 * 3.14159 * 78.75 m² Curved Surface Area = 6.28318 * 78.75 m² Curved Surface Area ≈ 494.80 m²

So, the inner curved surface area of the well is approximately 494.80 square meters. Now we need to find the cost of plastering this area. We're given that the cost is ₹3 per square meter. Therefore, the total cost of plastering is:

Total Cost = Curved Surface Area * Cost per square meter Total Cost = 494.80 m² * ₹3/m² Total Cost ≈ ₹1484.40

Therefore, the cost of plastering the inner curved surface of the well is approximately ₹1484.40. That's the final piece of the puzzle!

Importance of Surface Area Calculation

The calculation of the surface area of the well's interior is just as critical as determining the volume of earth to be excavated. This measurement directly influences the amount of plaster required for the lining, which in turn dictates the material costs. Accurate surface area estimation prevents underestimation, which could lead to insufficient plaster, or overestimation, which results in unnecessary expenditure.

Moreover, the surface area calculation is essential for estimating the labor needed for plastering. The larger the surface area, the more time and effort required to apply the plaster evenly and effectively. A precise surface area calculation enables the contractor to allocate the appropriate number of workers and estimate the duration of the plastering task. This helps in scheduling the project efficiently and avoiding delays.

Furthermore, the surface area calculation is crucial for ensuring the structural integrity and longevity of the well. Proper plastering protects the well's walls from water damage, soil erosion, and other environmental factors that can compromise its stability. An accurate surface area calculation ensures that the plaster is applied uniformly and adequately, providing a robust barrier against these threats.

In addition to these practical considerations, the surface area calculation is also important for aesthetic purposes. A well-plastered interior not only enhances the well's functionality but also improves its visual appeal. An accurate surface area calculation ensures that the plaster is applied smoothly and evenly, creating a clean and attractive finish. This can be particularly important in wells that are located in public areas or are used for decorative purposes.

In conclusion, determining the surface area of the well's interior is not just a matter of mathematical calculation; it is a crucial step in ensuring the well's structural integrity, longevity, and aesthetic appeal. Accurate surface area estimation is essential for efficient resource allocation, effective labor management, and the overall success of the well construction project.

Summary

So, to recap:

  • The amount of earth that must be dug out is approximately 865.90 cubic meters.
  • The cost of plastering the inner curved surface is approximately ₹1484.40.

Easy peasy, right? These calculations are super useful in real-world construction and give you a solid understanding of the math behind such projects. Now you can confidently tackle similar problems. Keep practicing, and you'll become a pro in no time!

Final Thoughts

Understanding the math behind practical problems like digging a well is super important. It's not just about plugging numbers into formulas; it's about understanding the real-world implications of those numbers. The volume of earth you dig out affects everything from labor costs to environmental impact, and the surface area you plaster affects the well's structural integrity and longevity. By mastering these calculations, you're not just solving math problems; you're gaining valuable skills that can be applied in a variety of fields.

So, next time you see a well, remember the math that went into creating it! It's a great example of how math is all around us, helping us build and understand the world we live in. And remember, practice makes perfect. The more you work with these types of problems, the more comfortable and confident you'll become. Keep exploring, keep learning, and keep applying math to the world around you!