Cylinder Volume Calculation When Height Is Twice The Radius

The question at hand involves finding the volume of a cylinder, a fundamental concept in geometry. Cylinders are three-dimensional shapes characterized by two parallel circular bases connected by a curved surface. Understanding how to calculate their volume is essential in various fields, from engineering and physics to everyday applications like calculating the capacity of tanks or containers. To tackle this problem effectively, we need to recall the formula for the volume of a cylinder and how it relates to the cylinder's dimensions: radius and height. The problem specifically states that the height of the cylinder is twice the radius of its base, which introduces a crucial relationship we must incorporate into our calculations. We'll explore how to translate this relationship into a mathematical expression and then use it to determine the volume. The goal is not just to find the correct answer but also to understand the underlying principles and how they apply to this particular scenario. This involves carefully analyzing the given information, selecting the appropriate formula, and substituting the given relationship to arrive at the correct expression for the cylinder's volume. Let's dive into the details and break down the problem step by step to gain a clear understanding of the solution.

The Formula for Cylinder Volume: πr²h

To calculate the volume of a cylinder, we use the formula V = πr²h, where 'V' represents the volume, 'π' (pi) is a mathematical constant approximately equal to 3.14159, 'r' is the radius of the circular base, and 'h' is the height of the cylinder. This formula is derived from the basic principle that the volume of any prism (including a cylinder) is the area of its base multiplied by its height. In the case of a cylinder, the base is a circle, and the area of a circle is given by πr². Therefore, multiplying this area by the height 'h' gives us the volume of the cylinder. Understanding this formula is crucial for solving problems related to cylinder volumes. It allows us to relate the physical dimensions of the cylinder (radius and height) to its capacity (volume). The formula highlights the importance of both the radius and the height in determining the volume; a larger radius or a greater height will result in a larger volume. The presence of π in the formula reflects the circular nature of the cylinder's base, emphasizing the connection between circles and cylinders in geometry. Before we can apply this formula to our specific problem, we need to consider the given relationship between the height and the radius. The problem states that the height is twice the radius, which means we can express the height in terms of the radius, simplifying the volume calculation.

Relating Height and Radius: h = 2x

The problem states a crucial relationship: the height of the cylinder is twice the radius of its base. To translate this into a mathematical expression, let's represent the radius by the variable 'x'. Since the height is twice the radius, we can express the height 'h' as h = 2x. This simple equation is the key to solving the problem because it allows us to express the volume of the cylinder in terms of a single variable, 'x'. By substituting 2x for 'h' in the volume formula, we eliminate one variable and simplify the calculation. This step demonstrates the power of using algebraic expressions to represent geometric relationships. The equation h = 2x encapsulates the geometric constraint given in the problem statement, allowing us to incorporate this information into the volume calculation. It's important to note that this relationship holds true for all cylinders where the height is twice the radius, making it a general principle applicable to a variety of scenarios. Now that we have expressed both the radius and the height in terms of 'x', we can substitute these expressions into the volume formula and simplify to find the expression that represents the cylinder's volume. This substitution is the next key step in solving the problem, bringing us closer to the final answer.

Substituting into the Volume Formula: V = πx²(2x)

Now that we know the radius is 'x' and the height is '2x', we can substitute these values into the formula for the volume of a cylinder, V = πr²h. Replacing 'r' with 'x' and 'h' with '2x', we get V = π(x²)(2x). This substitution is a critical step because it expresses the volume of the cylinder entirely in terms of the variable 'x'. This makes it easier to simplify the expression and arrive at the correct answer. The substitution process demonstrates how algebraic manipulation can be used to solve geometric problems. By replacing the variables with their equivalent expressions, we transform the volume formula into a form that can be readily simplified. This step highlights the interconnectedness of algebra and geometry, showing how algebraic techniques can be applied to solve geometric problems. The resulting expression, V = πx²(2x), is now ready for simplification. We can use the rules of algebra to combine the terms and arrive at a more concise expression for the volume. This simplification process is the next key step in solving the problem, leading us closer to the final answer and allowing us to identify the correct option among the given choices.

Simplifying the Expression: V = 2πx³

After substituting the values into the volume formula, we have V = πx²(2x). To simplify this expression, we multiply the terms together. Multiplying πx² by 2x, we get V = 2πx³. This simplified expression represents the volume of the cylinder in terms of 'x', where 'x' is the radius of the base. The simplification process involves applying the basic rules of algebra, such as multiplying coefficients and adding exponents. In this case, we multiplied the coefficients (π and 2) and added the exponents of 'x' (2 and 1) to arrive at the simplified expression. This step demonstrates the importance of algebraic skills in solving mathematical problems, particularly in geometry. The simplified expression, V = 2πx³, is now in its most concise form, making it easy to identify the correct answer among the given options. This expression clearly shows the relationship between the volume of the cylinder and its radius, highlighting the cubic relationship between the volume and the radius. The factor of 2π indicates the constant proportionality between the volume and the cube of the radius, determined by the cylinder's geometry. Now that we have the simplified expression for the volume, we can compare it to the given options and select the correct answer.

Identifying the Correct Option: Option D

Having simplified the expression for the volume of the cylinder to V = 2πx³, we can now compare this result to the given options. The options are:

A. 4πx² B. 2πx² C. πx²+2x D. 2πx³

By comparing our simplified expression, V = 2πx³, with the options, we can clearly see that option D, 2πx³, matches our result. Therefore, option D is the correct answer. This final step demonstrates the importance of careful calculation and simplification in problem-solving. By following the steps methodically and applying the correct formulas and algebraic techniques, we have successfully arrived at the solution. The correct answer, 2πx³, represents the volume of the cylinder in cubic units, given the condition that the height is twice the radius. This solution highlights the practical application of geometric formulas and algebraic manipulation in solving real-world problems. It also reinforces the importance of understanding the relationships between different geometric properties and how they can be expressed mathematically. With the correct answer identified, we have successfully solved the problem and demonstrated a clear understanding of cylinder volume and its relationship to radius and height.

Conclusion: Understanding Cylinder Volume and Its Applications

In conclusion, by carefully applying the formula for cylinder volume and incorporating the given relationship between height and radius, we successfully determined that the expression representing the volume of the cylinder is 2πx³. This problem highlights the importance of understanding fundamental geometric concepts and their algebraic representations. The ability to translate word problems into mathematical expressions and then simplify those expressions is a crucial skill in mathematics and various related fields. Understanding cylinder volume has practical applications in numerous areas, including engineering, architecture, and manufacturing. For example, engineers use these calculations to design tanks and containers, architects use them to calculate the volume of cylindrical structures, and manufacturers use them to determine the amount of material needed to produce cylindrical objects. This problem also demonstrates the interconnectedness of different mathematical concepts. Geometry, algebra, and problem-solving skills are all essential for tackling this type of question. By mastering these concepts, students can develop a strong foundation in mathematics and be well-prepared for more advanced topics. The process of solving this problem, from understanding the formula to simplifying the expression and identifying the correct answer, reinforces the importance of a systematic approach to problem-solving. Each step builds upon the previous one, leading to a clear and logical solution. This approach can be applied to a wide range of mathematical problems, making it a valuable skill for students to develop. The solution to this problem not only provides the correct answer but also enhances our understanding of cylinder volume and its applications in the real world.