Calculate Painting Cost For Cylindrical Tank With Hemispherical Ends

In this article, we will delve into a practical problem involving the calculation of the cost to paint a storage tank. This tank has a unique design, comprising a circular cylinder with hemispherical ends. Understanding how to calculate surface areas of such composite shapes is crucial in various fields, from engineering to construction. The specific problem we'll address involves a tank with a diameter of 1.4 meters and a cylindrical length of 8 meters. Our goal is to determine the cost of painting the tank's exterior at a rate of Rs. 10 per square meter. This exercise will not only help us with this particular scenario but also enhance our skills in calculating surface areas and applying them to real-world cost estimations. The solution involves breaking down the tank into its basic geometric shapes, calculating the surface area of each part, summing them up to find the total surface area, and finally, calculating the painting cost. So, let's embark on this step-by-step journey to solve this interesting problem.

Problem Statement: Calculating the Painting Cost of a Storage Tank

Let's clearly define the problem we are going to solve. We have a storage tank that is shaped like a cylinder with hemispheres attached to both ends. The cylinder has a diameter of 1.4 meters and a length of 8 meters. The task is to find the total cost of painting the outside of this tank, given that the painting rate is Rs. 10 per square meter. To solve this, we need to calculate the total surface area to be painted. This involves finding the curved surface area of the cylinder and the surface areas of the two hemispheres. Once we have the total area, we can easily calculate the total painting cost by multiplying the area by the cost per square meter. This problem combines geometry and basic arithmetic, making it a great exercise in applying mathematical concepts to practical situations. In the following sections, we will break down the solution step by step, ensuring a clear and comprehensive understanding of each calculation involved.

Step 1: Identifying the Components of the Storage Tank

To accurately calculate the surface area of the storage tank, we first need to break it down into its individual geometric components. The tank is comprised of two main shapes: a circular cylinder and two hemispheres. These hemispheres are attached to either end of the cylinder. This composite structure is crucial to understand because each component has a unique formula for calculating its surface area. The cylinder contributes its curved surface area, while the hemispheres together form a sphere, contributing the sphere's surface area. By recognizing these distinct parts, we can apply the appropriate formulas and calculate the total surface area of the tank. This breakdown is a fundamental step in solving the problem and ensures that we account for all the surfaces that need to be painted. In the next steps, we will delve into the specific measurements and formulas needed for each component.

Detailed Description of the Cylindrical Part

The cylindrical part of the storage tank is a crucial component that contributes significantly to the overall surface area. To understand its contribution, let's describe it in detail. The cylinder has a diameter of 1.4 meters, which means its radius is half of that, or 0.7 meters. The length (or height) of the cylindrical part is given as 8 meters. When calculating the surface area, we are interested in the curved surface area of the cylinder, as the circular ends are covered by the hemispheres. The formula for the curved surface area of a cylinder is 2πrh, where 'r' is the radius and 'h' is the height (or length). By substituting the values we have, we can calculate the curved surface area of this part. This area represents the portion of the tank's surface that needs to be painted and is a vital part of our calculation for the total painting cost. Understanding the dimensions and the appropriate formula for the cylinder is essential for accurately solving the problem.

Detailed Description of the Hemispherical Ends

The two hemispherical ends of the storage tank play a significant role in determining the total surface area. Each end is a hemisphere, which is essentially half of a sphere. Since we have two hemispheres, their combined surface area is equivalent to the surface area of a full sphere. The diameter of each hemisphere is the same as the diameter of the cylinder, which is 1.4 meters. This means the radius of each hemisphere is 0.7 meters. The formula for the surface area of a sphere is 4πr², where 'r' is the radius. Therefore, to find the combined surface area of the two hemispheres, we can simply calculate the surface area of a sphere with a radius of 0.7 meters. This area represents another significant portion of the tank's surface that needs to be painted. Accurate calculation of the hemispheres' surface area is crucial for determining the total painting cost. By understanding the geometry of hemispheres and applying the correct formula, we ensure the accuracy of our final result.

Step 2: Calculating the Surface Area of the Cylinder

Now that we have identified the cylindrical part of the tank, let's calculate its surface area. As mentioned earlier, we are interested in the curved surface area of the cylinder because the circular ends are covered by the hemispheres. The formula for the curved surface area of a cylinder is 2πrh, where 'r' is the radius and 'h' is the height (or length) of the cylinder. In our case, the radius 'r' is 0.7 meters, and the length 'h' is 8 meters. Substituting these values into the formula, we get: 2 * π * 0.7 * 8. This calculation will give us the surface area of the cylindrical part in square meters. The result is a crucial component in determining the total surface area of the tank. By accurately calculating this area, we ensure the precision of our final cost estimation. Let's perform this calculation and proceed to the next step of finding the surface area of the hemispheres.

The calculation is as follows:

Curved Surface Area of Cylinder = 2πrh

= 2 * (22/7) * 0.7 m * 8 m

= 2 * 22 * 0.1 m * 8 m

= 35.2 m²

Therefore, the curved surface area of the cylindrical part of the tank is 35.2 square meters.

Step 3: Calculating the Surface Area of the Hemispheres

Next, we need to calculate the surface area of the hemispherical ends of the tank. As we discussed, the two hemispheres together form a complete sphere. Therefore, we can calculate the surface area of a sphere with the same radius as the hemispheres. The formula for the surface area of a sphere is 4πr², where 'r' is the radius. In our case, the radius 'r' is 0.7 meters. Substituting this value into the formula, we get: 4 * π * (0.7)². This calculation will give us the combined surface area of the two hemispheres in square meters. This area is another significant component in determining the total surface area of the tank. Accurate calculation of this area is crucial for the overall precision of our cost estimation. Let's proceed with this calculation to complete the surface area calculations.

The calculation is as follows:

Surface Area of Sphere = 4πr²

= 4 * (22/7) * (0.7 m)²

= 4 * (22/7) * 0.49 m²

= 4 * 22 * 0.07 m²

= 6.16 m²

Thus, the combined surface area of the two hemispheres is 6.16 square meters.

Step 4: Determining the Total Surface Area to be Painted

Now that we have calculated the surface areas of both the cylindrical part and the hemispherical ends, we can determine the total surface area that needs to be painted. To do this, we simply add the curved surface area of the cylinder and the combined surface area of the two hemispheres. From our previous calculations, we know that the curved surface area of the cylinder is 35.2 square meters, and the combined surface area of the hemispheres is 6.16 square meters. Adding these two values together will give us the total surface area to be painted. This total area is essential for calculating the overall cost of painting the tank. By accurately summing these areas, we ensure that our final cost estimation is as precise as possible. Let's add these values to find the total surface area.

The calculation is as follows:

Total Surface Area = Curved Surface Area of Cylinder + Surface Area of Hemispheres

= 35.2 m² + 6.16 m²

= 41.36 m²

Therefore, the total surface area to be painted is 41.36 square meters.

Step 5: Calculating the Total Cost of Painting

With the total surface area of the tank calculated, we can now determine the total cost of painting. We are given that the cost of painting is Rs. 10 per square meter. To find the total cost, we simply multiply the total surface area by the cost per square meter. From our previous calculations, the total surface area to be painted is 41.36 square meters. Multiplying this by Rs. 10 will give us the total cost of painting the entire tank. This final calculation provides the answer to our original problem, giving us a clear understanding of the expense involved in painting the storage tank. Let's perform this final calculation to complete our solution.

The calculation is as follows:

Total Cost of Painting = Total Surface Area * Cost per Square Meter

= 41.36 m² * Rs. 10/m²

= Rs. 413.6

Therefore, the total cost of painting the storage tank is Rs. 413.6.

Conclusion

In conclusion, we have successfully calculated the total cost of painting the storage tank by breaking down the problem into manageable steps. We identified the tank's components as a cylinder and two hemispheres, calculated the surface area of each part, summed them to find the total surface area, and finally, multiplied the total area by the cost per square meter to find the total painting cost. The final cost of painting the storage tank is Rs. 413.6. This exercise demonstrates the practical application of geometric formulas in real-world scenarios. Understanding how to calculate surface areas of composite shapes is crucial in various fields, including engineering, construction, and design. By mastering these calculations, we can accurately estimate costs and plan projects effectively. This problem-solving approach can be applied to various other situations, making it a valuable skill to develop. The step-by-step method we followed ensures clarity and accuracy in the solution, providing a comprehensive understanding of the process involved.