Well Plastering Cost: A Step-by-Step Calculation

by TextBrain Team 49 views

Hey guys! Ever wondered how to calculate the cost of plastering the inside of a well? It's actually a pretty common problem in math, and today, we're going to break it down step by step. We'll take a look at a typical well problem and show you exactly how to figure out the plastering cost. So, grab your calculators, and let's dive in!

Understanding the Problem

Before we jump into the calculations, let's make sure we understand what the problem is asking. We're dealing with a cylindrical well, and we need to find the cost of plastering its inner curved surface. This means we're not plastering the top or the bottom – just the inside walls. The problem gives us two crucial pieces of information: the inner diameter of the well and its depth. We also know the cost per square meter for plastering.

Key Information

  • Inner Diameter: 3.5 meters
  • Depth (Height): 10 meters
  • Plastering Rate: 40 rupees per square meter

Our goal is to find the total cost of plastering the inner curved surface.

Step 1: Find the Radius

The diameter is the distance across the circle through the center. The radius, on the other hand, is the distance from the center to any point on the circle. To find the radius, we simply divide the diameter by 2. In our case:

Radius = Diameter / 2 = 3.5 meters / 2 = 1.75 meters

So, the radius of our well is 1.75 meters. Keep this value handy, as we'll need it for the next step.

Step 2: Calculate the Curved Surface Area

The curved surface area (CSA) of a cylinder is the area of its side, not including the top and bottom. The formula for the CSA of a cylinder is:

CSA = 2 * π * r * h

Where:

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

Now, let's plug in the values we have:

CSA = 2 * 3.14159 * 1.75 meters * 10 meters CSA = 2 * 3.14159 * 17.5 square meters CSA ≈ 109.956 square meters

So, the inner curved surface area of the well is approximately 109.956 square meters. This is the area we need to plaster.

Step 3: Calculate the Total Cost

Now that we know the area to be plastered and the cost per square meter, we can easily find the total cost. We simply multiply the area by the plastering rate:

Total Cost = CSA * Plastering Rate Total Cost = 109.956 square meters * 40 rupees per square meter Total Cost ≈ 4398.24 rupees

Therefore, the cost of plastering the inner curved surface of the well is approximately 4398.24 rupees. And that's it! We've successfully calculated the plastering cost.

Breaking Down the Formula and Concepts

Let's spend some more time exploring the concepts so you can really nail similar problems. Understanding why the formulas work is just as important as knowing how to use them!

Cylinders and Their Properties

A cylinder is a 3D geometric shape with two parallel circular bases connected by a curved surface. Think of a can of soup or a toilet paper roll. The key properties we need to worry about here are the radius (r) and the height (h).

Understanding Surface Area

Surface area is the total area of the outer surface of a 3D object. For a cylinder, we have three main surface areas to consider:

  • Top Surface: A circle with area Ï€r²
  • Bottom Surface: Another circle with area Ï€r²
  • Curved Surface: The area we calculated earlier, 2Ï€rh

The total surface area of a closed cylinder (including the top and bottom) would be 2πr² + 2πrh. However, in our well problem, we only cared about the curved surface area.

Why is the Curved Surface Area Formula 2Ï€rh?

Imagine cutting the curved surface of the cylinder vertically and unrolling it. What shape do you get? You get a rectangle! The height of the rectangle is the same as the height of the cylinder (h). The width of the rectangle is the circumference of the circular base, which is 2Ï€r. Therefore, the area of the rectangle (and the curved surface area of the cylinder) is length * width = 2Ï€r * h = 2Ï€rh.

Common Mistakes to Avoid

When tackling these types of problems, it's easy to make a few common mistakes. Here are some things to watch out for:

  • Using the Diameter Instead of the Radius: This is a classic mistake. Remember, the formulas use the radius, not the diameter. If you're given the diameter, make sure to divide it by 2 to get the radius first.
  • Forgetting Units: Always include the units in your calculations and final answer. In this case, we're dealing with meters and square meters.
  • Incorrectly Applying the Formula: Make sure you're using the correct formula for the curved surface area. Double-check that you're plugging in the values correctly.
  • Not Reading the Problem Carefully: Always read the problem carefully to understand exactly what it's asking. In our case, we only needed to plaster the inner curved surface, not the entire well.
  • Rounding Errors: Be mindful of rounding errors, especially when dealing with Ï€. It's generally best to use the full value of Ï€ on your calculator until the very end of the calculation.

Practice Problems

Okay, guys, let's put your newfound knowledge to the test! Here are a couple of practice problems for you to try:

Practice Problem 1

A cylindrical tank has an inner diameter of 4.2 meters and is 12 meters deep. Find the cost of painting the inner curved surface at a rate of 50 rupees per square meter.

Practice Problem 2

A well has a radius of 2 meters and a depth of 8 meters. If the plastering cost is 45 rupees per square meter, what is the total cost of plastering the inner curved surface?

Try solving these problems on your own, and then check your answers. The steps are the same as the example we worked through earlier. Good luck!

Real-World Applications

Understanding how to calculate surface areas and costs has many practical applications in the real world. Here are just a few examples:

  • Construction: Estimating the amount of material needed for painting, plastering, or tiling.
  • Manufacturing: Calculating the amount of material needed to produce cylindrical containers or pipes.
  • Engineering: Designing tanks, silos, and other cylindrical structures.
  • Home Improvement: Figuring out how much paint you need to paint a cylindrical pillar or how much wallpaper to cover a curved wall.

These skills are valuable in various fields and can help you make informed decisions in everyday life.

Conclusion

So, there you have it! Calculating the cost of plastering the inner curved surface of a well isn't so mysterious after all. By breaking down the problem into smaller steps and understanding the underlying formulas, you can easily solve these types of problems. Remember to pay attention to the details, avoid common mistakes, and practice regularly. With a little bit of effort, you'll be a surface area master in no time! Keep practicing, and you'll be able to tackle any math problem that comes your way. You got this!