Calculating The Height Of Cylindrical Cans A Mathematical Approach
Uma indústria produz latas cilíndricas retas com tampa superior e base inferior. Sabendo que cada lata possui uma área total de superfície de 351,68 cm² e que o raio de sua base mede 4 cm, qual é a altura de cada lata?
In this comprehensive article, we delve into a fascinating geometrical problem involving cylindrical cans. Our focus centers on determining the height of these cans, given their total surface area and base radius. This exploration combines mathematical principles with practical applications, offering a unique perspective on the dimensions of everyday objects. As we journey through the intricacies of surface area calculations and algebraic manipulations, we will not only unravel the solution to this specific problem but also gain a deeper understanding of the underlying mathematical concepts. Join us as we embark on this enlightening expedition, where mathematical precision meets real-world relevance. This detailed analysis aims to provide clarity and insight into the process of solving geometric problems, enhancing your comprehension and analytical skills.
Problem Statement Unpacking the Cylindrical Can Challenge
Our primary objective is to calculate the height of cylindrical cans manufactured in a specific industrial setting. Each can is designed with a top lid and a bottom base, forming a complete cylindrical structure. We are provided with two crucial pieces of information: the total surface area of each can, which is 351.68 cm², and the radius of the circular base, which measures 4 cm. With these parameters, we embark on a mathematical quest to determine the unknown height of the cans. This problem encapsulates the essence of geometrical problem-solving, requiring a meticulous application of formulas and a clear understanding of spatial relationships. By dissecting the problem statement and identifying the given variables, we lay the foundation for a systematic approach to finding the solution. This initial step is crucial in any mathematical endeavor, setting the stage for accurate calculations and logical deductions. Understanding the problem fully is paramount to arriving at a correct and meaningful answer. This exploration into cylindrical dimensions offers a practical application of geometric principles, highlighting the relevance of mathematics in everyday scenarios.
Mathematical Foundation Laying the Groundwork for Calculations
To embark on our journey to solve this intriguing problem, we must first establish a solid mathematical foundation. Our primary tool in this endeavor is the formula for the total surface area of a cylinder. This formula encapsulates the geometric properties of a cylinder, providing a direct link between its dimensions and its surface area. Specifically, the total surface area (TSA) of a cylinder is given by the equation: TSA = 2πr² + 2πrh, where 'r' represents the radius of the base and 'h' denotes the height of the cylinder. This equation is not just a mere collection of symbols; it is a concise representation of the geometrical relationships inherent in a cylinder. The term 2πr² accounts for the combined area of the top and bottom circular faces, while the term 2πrh represents the lateral surface area, which is essentially the curved surface of the cylinder. Understanding this formula is paramount, as it forms the cornerstone of our problem-solving approach. By dissecting the formula and recognizing the significance of each component, we gain a deeper appreciation for the mathematical elegance underlying cylindrical geometry. This foundation allows us to translate the problem statement into a concrete mathematical equation, paving the way for accurate calculations and a precise determination of the cylinder's height. The careful application of this formula, coupled with our understanding of the given parameters, will lead us to the solution we seek. The mathematical groundwork laid here is not only essential for this specific problem but also serves as a valuable tool for tackling a wide range of geometrical challenges.
Applying the Formula A Step-by-Step Calculation
With the formula for the total surface area of a cylinder firmly in place (TSA = 2πr² + 2πrh), we now proceed to apply it to our specific problem. We are given that the total surface area (TSA) is 351.68 cm² and the radius (r) is 4 cm. Our mission is to find the height (h). The first step involves substituting the known values into the formula: 351.68 = 2π(4)² + 2π(4)h. This substitution transforms the abstract formula into a concrete equation tailored to our problem. Next, we simplify the equation by performing the necessary calculations. Squaring the radius gives us 4² = 16, and multiplying by 2π yields 32π. The equation now looks like this: 351.68 = 32π + 8πh. To isolate the term containing 'h', we subtract 32π from both sides of the equation: 351.68 - 32π = 8πh. Now, we must determine the numerical value of 32π. Using the approximation π ≈ 3.14, we find that 32π ≈ 100.48. Substituting this value into the equation gives us: 351.68 - 100.48 = 8πh, which simplifies to 251.2 = 8πh. To finally solve for 'h', we divide both sides of the equation by 8π: h = 251.2 / (8π). Again, using the approximation π ≈ 3.14, we calculate 8π ≈ 25.12. Therefore, h = 251.2 / 25.12, which simplifies to h = 10. Thus, the height of each can is 10 cm. This step-by-step calculation demonstrates the methodical application of the surface area formula, showcasing the precision required in mathematical problem-solving. The journey from the initial equation to the final answer highlights the power of algebraic manipulation and the importance of accurate arithmetic. This process not only solves the problem at hand but also reinforces our understanding of cylindrical geometry and equation-solving techniques.
Solution and Verification Confirming the Height of the Can
After meticulously applying the formula for the total surface area of a cylinder and performing the necessary calculations, we have arrived at a solution: the height (h) of each can is 10 cm. This is a significant result, but our journey doesn't end here. To ensure the accuracy of our solution, it is crucial to verify it. Verification is a cornerstone of mathematical rigor, providing confidence in our answer and highlighting any potential errors in our calculations. To verify our solution, we substitute the values we have—the radius (r) of 4 cm and the calculated height (h) of 10 cm—back into the original formula for the total surface area: TSA = 2πr² + 2πrh. Plugging in the values, we get: TSA = 2π(4)² + 2π(4)(10). Simplifying this equation, we have: TSA = 32π + 80π, which further simplifies to TSA = 112π. Now, we approximate π as 3.14 and calculate 112π: 112 * 3.14 = 351.68. This result matches the given total surface area of 351.68 cm², confirming the accuracy of our calculated height. The verification process not only validates our solution but also reinforces our understanding of the relationship between the dimensions of a cylinder and its surface area. This step is a testament to the importance of thoroughness in mathematical problem-solving, ensuring that our conclusions are both accurate and reliable. The successful verification of our solution solidifies our confidence in the calculated height of 10 cm, completing our exploration of this cylindrical can challenge.
Conclusion Reflecting on Cylindrical Dimensions
In conclusion, our journey to determine the height of the cylindrical cans has been a rewarding exploration of geometric principles and mathematical problem-solving. By carefully applying the formula for the total surface area of a cylinder and systematically working through the calculations, we successfully found the height of each can to be 10 cm. The verification process further solidified our confidence in this result, demonstrating the importance of thoroughness and accuracy in mathematical endeavors. This exercise has not only provided a solution to a specific problem but also enhanced our understanding of cylindrical geometry and the relationships between dimensions and surface area. The mathematical concepts we've employed here have broad applications in various fields, from engineering and design to everyday problem-solving. The ability to analyze and manipulate geometric formulas is a valuable skill, and this exploration has provided a practical context for developing that skill. As we reflect on our journey, we recognize the power of mathematics in unraveling the complexities of the world around us. Cylindrical shapes are ubiquitous, and understanding their properties is essential for a wide range of applications. This article has served as a testament to the elegance and utility of mathematical principles in addressing real-world challenges. The successful resolution of this problem underscores the importance of a solid foundation in mathematics and the value of a methodical approach to problem-solving. We hope this exploration has not only enlightened you on the specific dimensions of cylindrical cans but also inspired you to appreciate the beauty and power of mathematics in everyday life.