Calculate Painting Cost Of Open Wooden Box A Math Problem
Introduction
In this intricate mathematical problem, we delve into the realm of mensuration and cost estimation. Our task is to determine the total cost of painting the inner surface of an open wooden box. This box, crafted from sturdy wood 5 cm thick, boasts external dimensions of 2.55 m in length, 2.15 m in width, and 1.05 m in height. The painting cost is pegged at a rate of ₹100 per square meter. To unravel this problem, we must meticulously calculate the inner surface area of the box and then apply the given painting rate. This exploration will not only test our understanding of geometric principles but also hone our ability to apply these concepts to real-world scenarios. The problem highlights the practical application of mathematics in everyday situations, from home improvement projects to large-scale construction endeavors. Understanding how to calculate surface areas and costs is crucial for anyone involved in such activities, ensuring accurate budgeting and efficient resource management. Let's embark on this mathematical journey and uncover the solution step by step.
Dissecting the Dimensions: External vs. Internal
To accurately compute the cost, the first crucial step involves dissecting the dimensions of the open wooden box. We are provided with the external dimensions, which serve as our starting point. However, it's the internal dimensions that dictate the surface area to be painted. Given the wood's thickness of 5 cm, we must carefully subtract this thickness from the external dimensions to arrive at the internal ones. This process requires meticulous attention to detail and a clear understanding of how the thickness affects each dimension. For the length and width, the thickness is subtracted twice (once from each side), while for the height, it's subtracted only once since the box is open at the top. This distinction is crucial and often a source of error if overlooked. The conversion between meters and centimeters also plays a pivotal role in this step, ensuring consistent units throughout the calculation. By accurately determining the internal dimensions, we lay the foundation for calculating the inner surface area, a critical component in finding the total painting cost. This meticulous approach highlights the importance of precision in mathematical calculations, especially when dealing with real-world applications where errors can lead to significant discrepancies.
Calculating the Inner Surface Area
With the internal dimensions meticulously calculated, we now pivot to the core of the problem: determining the inner surface area of the open box. This involves a careful consideration of the box's geometry. Being open at the top, the box comprises a base and four sides. Each of these surfaces contributes to the total area to be painted, and their individual areas must be calculated and summed up. The base, a rectangle, has an area equal to the product of its internal length and width. The four sides, also rectangles, have areas determined by multiplying the internal height by the internal length and width, respectively. It's crucial to recognize that two pairs of sides will have identical areas, simplifying the calculation. The summation of these individual areas yields the total inner surface area, expressed in square meters. This value is the key to unlocking the final cost calculation. The process of calculating the inner surface area reinforces the application of geometric principles in practical scenarios. It showcases how understanding shapes and their properties allows us to solve real-world problems, from estimating painting needs to designing structures.
The Cost Calculation: Applying the Rate
Having meticulously determined the inner surface area of the open box, we now approach the final stage: calculating the total cost of painting. This involves a straightforward application of the given painting rate, which is ₹100 per square meter. The total cost is simply the product of the inner surface area (in square meters) and the painting rate. This calculation underscores the direct relationship between area and cost, a fundamental concept in many practical applications. The result, expressed in Indian Rupees (₹), represents the total expenditure required to paint the inner surface of the box. This final step brings the entire problem together, demonstrating how individual calculations culminate in a meaningful and practical solution. The cost calculation highlights the importance of understanding unit conversions and how they impact the final result. Ensuring that all values are in consistent units is crucial for accurate calculations, preventing errors that could lead to significant financial misestimates. This emphasizes the need for precision and attention to detail in all stages of the problem-solving process.
Summary of Steps
To recap, we embarked on a journey to calculate the total cost of painting the inner surface of an open wooden box. Our journey involved a series of meticulously executed steps: First, we dissected the dimensions, distinguishing between external and internal measurements, accounting for the wood's thickness. This step was crucial in accurately determining the space to be painted. Next, we calculated the inner surface area by summing the areas of the base and four sides, each computed using the internal dimensions. This step showcased the application of geometric principles in a practical context. Finally, we applied the given painting rate to the inner surface area to arrive at the total cost. This final step brought together all the preceding calculations to provide a tangible solution. Each step in this process underscored the importance of accuracy and attention to detail. From the initial dimension analysis to the final cost calculation, precision was paramount. The problem highlighted the interconnectedness of mathematical concepts and their applicability in real-world scenarios, reinforcing the value of mathematical literacy in everyday life.
Conclusion
In conclusion, this mathematical exploration has provided a comprehensive understanding of how to calculate the cost of painting the inner surface of an open wooden box. By meticulously dissecting the dimensions, computing the inner surface area, and applying the painting rate, we have successfully navigated the problem. This exercise not only reinforces our understanding of geometric principles but also highlights the practical application of mathematics in real-world scenarios. The problem serves as a testament to the importance of precision and attention to detail in mathematical calculations. From home improvement projects to large-scale construction endeavors, the ability to accurately estimate costs is crucial for effective planning and resource management. This exploration underscores the value of mathematical literacy and its role in empowering individuals to solve practical problems with confidence. The systematic approach employed throughout this exercise can be applied to a wide range of similar problems, demonstrating the versatility and power of mathematical thinking.
Problem
An open box is made of wood 5 cm thick. Its external dimensions are 2.55 m, 2.15 m and 1.05 m. What is the total cost of painting the inner surface of the box at ₹100 per square meter?
