Calculate Cylinder Volume When Height Is Twice The Radius

When dealing with geometric shapes, calculating the volume is a fundamental concept. In this article, we will delve into the specifics of finding the volume of a cylinder where there's a unique relationship between its height and radius: the height is twice the radius of its base. This problem not only reinforces our understanding of volume calculation but also helps us appreciate how geometric dimensions relate to each other. So, let's explore this fascinating geometric challenge and discover the correct expression for the volume of the cylinder.

Understanding the Cylinder and Its Properties

Before diving into the solution, it's essential to have a clear picture of what a cylinder is and its key properties. A cylinder is a three-dimensional geometric shape with two parallel circular bases connected by a curved surface. Imagine a can of soup or a roll of paper towels – these are everyday examples of cylinders. The two main dimensions of a cylinder that we need to consider when calculating its volume are the radius (*r*) of the circular base and the height (*h*) of the cylinder. The radius is the distance from the center of the circle to any point on its circumference, and the height is the perpendicular distance between the two circular bases.

The Formula for the Volume of a Cylinder

The volume (*V*) of a cylinder is determined by the space it occupies. To calculate this, we use a well-established formula that relates the radius (*r*) and the height (*h*) of the cylinder. The formula is:

${ V = \pi r^2 h }$

Here, π (pi) is a mathematical constant approximately equal to 3.14159. The term *r*^2 represents the area of the circular base, and multiplying this by the height *h* gives us the volume of the entire cylinder. This formula is a cornerstone in understanding and calculating the volumes of cylindrical objects, and it's crucial for solving a variety of real-world problems, from engineering designs to simple household measurements.

Problem Setup: Height as Twice the Radius

Now, let’s introduce the specific condition given in our problem: the height of the cylinder is twice the radius of its base. This relationship can be expressed mathematically as:

${ h = 2r }$

This simple equation adds an interesting twist to the volume calculation. Instead of treating the height and radius as independent variables, we now have a direct connection between them. This means that if we know the radius, we automatically know the height, and vice versa. This relationship is key to simplifying our volume formula and arriving at the correct expression. It’s a classic example of how constraints in geometric problems can lead to elegant solutions.

Substituting the Relationship into the Volume Formula

To find the expression for the volume of the cylinder under the given condition, we need to substitute the relationship ${ h = 2r }$ into the volume formula ${ V = \pi r^2 h }$. This substitution will allow us to express the volume in terms of a single variable, which will simplify our calculations.

By replacing *h* with *2r* in the volume formula, we get:

${ V = \pi r^2 (2r) }$

This step is crucial because it transforms the volume formula from being dependent on two variables (*r* and *h*) to being dependent on just one (*r*). This makes the expression easier to work with and allows us to find the correct answer among the given options. The substitution is a powerful technique in mathematics, allowing us to solve problems more efficiently by reducing the number of variables we need to consider.

Simplifying the Expression

Now that we have substituted the relationship between the height and the radius into the volume formula, the next step is to simplify the expression. This involves basic algebraic manipulation to combine like terms and present the formula in its most concise form. Starting with our substituted formula:

${ V = \pi r^2 (2r) }$

We can multiply the terms together. Remember that when multiplying terms with exponents, we add the exponents if the bases are the same. In this case, we have *r*^2 multiplied by *r*, which is the same as *r*^1. So, we add the exponents 2 and 1 to get 3.

${ V = 2 \pi r^3 }$

This simplified expression, ${ V = 2 \pi r^3 }$, is the key to solving our problem. It tells us that the volume of the cylinder is directly proportional to the cube of its radius. This means that if we double the radius, the volume will increase by a factor of eight (2 cubed). The simplified formula not only makes the calculation easier but also provides a clearer understanding of how the volume changes with the radius.

Identifying the Correct Option

Our simplified expression for the volume of the cylinder is ${ V = 2 \pi r^3 }$. To match this with the given options, we need to recognize that the problem uses *x* to represent the radius (*r*). So, we simply replace *r* with *x* in our formula:

${ V = 2 \pi x^3 }$

Now, we can compare this expression with the given options:

A. ${ 4 \pi x^2 }$

B. ${ 2 \pi x^3 }$

C. ${ \pi x^2 + 2x }$

D. ${ 2 + \pi x^3 }$

By comparing our derived expression with the options, it becomes clear that option B, ${ 2 \pi x^3 }$, is the correct one. This step highlights the importance of careful algebraic manipulation and substitution to arrive at the final answer. It also demonstrates how understanding the underlying mathematical principles can lead to a straightforward solution.

Why Other Options Are Incorrect

To fully understand the solution, it's helpful to examine why the other options are incorrect. This not only reinforces the correct methodology but also helps in avoiding common mistakes in similar problems.

Option A: ${ 4 \pi x^2 }$

This option looks similar to the correct answer but has a different exponent for *x*. The expression ${ x^2 }$ suggests that this might represent an area rather than a volume, as volume requires a cubic unit (${ x^3 }$). The coefficient 4 is also incorrect, as it doesn't match the correct coefficient of 2 in our derived formula.

Option C: ${ \pi x^2 + 2x }$

This option is a combination of terms with different dimensions. The term ${ \pi x^2 }$ represents the area of the base of the cylinder, while ${ 2x }$ represents a linear measurement. Adding these together doesn't make sense in the context of volume calculation, as we cannot add areas and lengths to get a volume.

Option D: ${ 2 + \pi x^3 }$

This option has an addition of a constant (2) to a term involving ${ \pi x^3 }$. This form is incorrect because the volume should be directly proportional to ${ x^3 }$ without any added constants. The constant 2 here doesn't have a clear geometric interpretation in the context of the cylinder's volume.

By understanding why these options are incorrect, we gain a deeper appreciation of the correct solution and the principles behind it. This critical analysis is a valuable skill in mathematics and problem-solving in general.

Conclusion

In conclusion, the expression that represents the volume of a cylinder where the height is twice the radius of its base is ${ 2 \pi x^3 }$, which corresponds to option B. This solution was derived by understanding the relationship between the height and radius, substituting this relationship into the volume formula, and simplifying the resulting expression. The process highlights the importance of understanding the fundamental formulas, algebraic manipulation, and dimensional analysis in solving geometric problems. By mastering these concepts, we can confidently tackle a wide range of mathematical challenges.

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