Determining The Final Volume Of A Cylinder Using Limits

In the realm of calculus and mathematical analysis, understanding how quantities change over time is a fundamental concept. This exploration delves into a specific scenario involving the volume of a cylinder, where the volume, denoted as V(t), varies with respect to time (t) according to the expression V(t) = (t+2)/√(9t²+1). Our primary objective is to determine the final volume of the cylinder as time progresses indefinitely. This problem isn't merely an exercise in applying mathematical formulas; it's a journey into the heart of limits, asymptotic behavior, and the physical interpretation of mathematical models. To find the final volume, we need to analyze the behavior of V(t) as t approaches infinity. This involves techniques from calculus, such as evaluating limits of functions, and requires a careful understanding of how the numerator and denominator of the expression interact as t grows larger. This analysis is crucial in various fields, including engineering, physics, and computer science, where understanding the long-term behavior of systems is essential for design and prediction. The problem challenges us to think critically about the relationship between mathematical models and real-world phenomena. The final volume isn't just a number; it represents a state the cylinder's volume approaches over an extended period. This could be a maximum capacity, a stable equilibrium, or even a point of diminishing returns. Thus, accurately determining the final volume allows us to make informed predictions and draw meaningful conclusions about the system being modeled. Furthermore, this exploration touches upon the core principles of calculus and mathematical analysis. It reinforces our understanding of limits, infinity, and the behavior of functions. By working through the problem, we strengthen our analytical skills, problem-solving abilities, and the capacity to connect abstract mathematical concepts with practical applications. The journey to find the final volume is not just about arriving at an answer; it's about the process of exploration, the application of mathematical tools, and the deeper understanding we gain along the way. So, let's embark on this mathematical journey and unravel the mystery of the cylinder's final volume.

Understanding the Volume Function V(t)

The cornerstone of our analysis is the volume function itself: V(t) = (t+2)/√(9t²+1). This expression encapsulates the relationship between the cylinder's volume and time. To effectively determine the final volume, we need to dissect this function, understand its components, and how they interact. The function consists of a numerator, (t+2), and a denominator, √(9t²+1). The numerator represents a linear function, increasing steadily as time (t) increases. The denominator, on the other hand, involves a square root of a quadratic expression. As t grows larger, the term 9t² dominates the behavior of the denominator. This interplay between the linear growth of the numerator and the square root growth of the denominator is what ultimately dictates the long-term behavior of V(t). Understanding the behavior of this function requires a strong foundation in mathematical concepts, particularly limits. A limit describes the value that a function approaches as the input (in this case, time) gets closer and closer to a specific value (in this case, infinity). To find the final volume, we need to evaluate the limit of V(t) as t approaches infinity. This involves techniques for simplifying the expression, identifying dominant terms, and applying limit laws. Furthermore, it's important to consider the physical context of the problem. The volume of a cylinder cannot be negative, and it's unlikely to grow without bound in a real-world scenario. Therefore, we expect the limit of V(t) as t approaches infinity to be a finite, non-negative value. This physical intuition can guide our mathematical analysis and help us interpret the results. Analyzing the function also involves considering its domain and any potential discontinuities. In this case, the function is defined for all non-negative values of t, since time cannot be negative. The denominator √(9t²+1) is always positive, so there are no points of discontinuity. This ensures that the function is well-behaved and we can focus on its asymptotic behavior as t approaches infinity. The process of understanding V(t) is not just about manipulating the mathematical expression; it's about building a mental model of how the cylinder's volume changes over time. This involves visualizing the graph of the function, considering its rate of change, and identifying any key features or trends. By developing this comprehensive understanding, we can approach the problem of finding the final volume with confidence and insight.

Evaluating the Limit as t Approaches Infinity

The crucial step in determining the final volume is evaluating the limit of V(t) as t approaches infinity: lim (t→∞) V(t) = lim (t→∞) (t+2)/√(9t²+1). This limit represents the value that V(t) approaches as time goes on indefinitely. To evaluate this limit, we need to employ techniques from calculus that allow us to handle expressions involving infinity. A common approach is to divide both the numerator and the denominator by the highest power of t present in the denominator. In this case, the highest power of t inside the square root is t², so we effectively divide by t. However, since t is inside the square root in the denominator, we divide by t in the numerator and √(t²) in the denominator. This gives us: lim (t→∞) (t+2)/√(9t²+1) = lim (t→∞) [(t+2)/t] / [√(9t²+1)/√(t²)]. Simplifying further, we get: lim (t→∞) (1 + 2/t) / √(9 + 1/t²). Now, as t approaches infinity, the terms 2/t and 1/t² approach zero. This is because any constant divided by a very large number becomes very small. Applying this knowledge, we can rewrite the limit as: lim (t→∞) (1 + 2/t) / √(9 + 1/t²) = (1 + 0) / √(9 + 0) = 1 / √9 = 1/3. Therefore, the limit of V(t) as t approaches infinity is 1/3. This result is significant because it tells us the final volume of the cylinder. It's not growing indefinitely, nor is it shrinking to zero. Instead, it's approaching a specific value, 1/3. This value represents the long-term equilibrium volume of the cylinder. The process of evaluating the limit involved several key steps: identifying the dominant terms, dividing by the highest power of t, and applying limit laws. Each of these steps is essential for accurately determining the limit and understanding the behavior of the function as t approaches infinity. It's also important to note that the limit provides valuable information about the physical system being modeled. In this case, it tells us that the cylinder's volume will eventually stabilize at a value of 1/3. This could be due to various factors, such as constraints on the inflow or outflow of fluid, or a balance between competing forces. The limit, therefore, is not just a mathematical result; it's a piece of the puzzle in understanding the real-world behavior of the cylinder.

Interpreting the Result and Final Answer

Having meticulously evaluated the limit of V(t) as t approaches infinity, we've arrived at a crucial result: lim (t→∞) V(t) = 1/3. But what does this 1/3 signify in the context of the cylinder's volume? This value represents the final volume that the cylinder approaches as time progresses indefinitely. It's not necessarily the maximum volume the cylinder ever reaches, but rather the volume it settles towards over a long period. This understanding is paramount in interpreting the result accurately. The result, 1/3, directly contradicts options A and B, which suggest the volume either grows or shrinks indefinitely. The limit exists and is finite, meaning the volume stabilizes over time, rather than diverging to infinity or zero. This finite limit indicates a state of equilibrium or a long-term stable condition for the cylinder's volume. The cylinder's volume is neither unbounded nor diminishing to nothingness; it's converging towards a specific, non-zero value. Therefore, the correct interpretation is that the volume approaches 1/3 as time tends to infinity. This understanding can be further contextualized by considering potential real-world scenarios. Perhaps the cylinder is part of a system where inflow and outflow rates eventually balance, leading to a stable volume. Or, there might be physical constraints on the cylinder's size, preventing it from expanding beyond a certain limit. The mathematical result, 1/3, provides a quantitative measure of this stability, offering insights into the system's long-term behavior. The interpretation of this result extends beyond mere numerical value. It involves translating the mathematical outcome into a meaningful statement about the physical system being modeled. We've not only calculated the limit but also deciphered its implications, linking abstract mathematics to concrete reality. This interpretive step is critical in problem-solving and decision-making, allowing us to draw conclusions, make predictions, and gain a deeper understanding of the processes at play. Therefore, the final answer is that the volume of the cylinder approaches 1/3 as time tends to infinity, demonstrating a stable and predictable long-term behavior.

Our journey to determine the final volume of the cylinder, described by the function V(t) = (t+2)/√(9t²+1), has been a rewarding exploration of calculus and its applications. We began by understanding the function itself, dissecting its components, and recognizing the interplay between the numerator and denominator. This laid the foundation for our critical step: evaluating the limit of V(t) as t approaches infinity. Through careful algebraic manipulation and the application of limit laws, we arrived at the result lim (t→∞) V(t) = 1/3. This result is not merely a mathematical outcome; it's a powerful statement about the long-term behavior of the cylinder's volume. It tells us that the volume neither grows indefinitely nor shrinks to zero. Instead, it approaches a stable, finite value of 1/3. This signifies a state of equilibrium or a long-term stable condition, providing valuable insights into the system being modeled. The process of solving this problem has reinforced several key concepts in calculus, including limits, infinity, and asymptotic behavior. We've seen how these abstract mathematical tools can be used to analyze real-world scenarios and make meaningful predictions. Furthermore, we've emphasized the importance of interpretation. The mathematical result, 1/3, gains its true significance when we understand its implications in the context of the problem. It's not just a number; it's a representation of the cylinder's final volume, a stable state that the system approaches over time. This exploration underscores the interconnectedness of mathematics and the physical world. Calculus provides a powerful framework for modeling dynamic systems, and the insights we gain from these models can have profound implications in various fields, from engineering and physics to economics and computer science. As we conclude this analysis, it's important to recognize that the journey of problem-solving is as valuable as the destination. The skills and understanding we've developed along the way – from analyzing functions and evaluating limits to interpreting results and drawing conclusions – will serve us well in future mathematical endeavors and beyond. The final volume of the cylinder, 1/3, stands as a testament to the power of calculus and the beauty of mathematical analysis.