Given The Volume Of A Cylinder As $4 \pi X^3$ Cubic Units And Its Height As $x$ Units, How Do You Express The Radius Of The Cylinder In Units?

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In the realm of geometry, understanding the relationships between different properties of shapes is crucial. One common shape we encounter is the cylinder, a three-dimensional object with two parallel circular bases connected by a curved surface. The volume of a cylinder represents the amount of space it occupies, while the radius is the distance from the center of the circular base to its edge, and the height is the perpendicular distance between the two bases. This article delves into a problem where we are given the volume and height of a cylinder and are tasked with finding an expression that represents its radius. This problem is a great exercise in applying the formula for the volume of a cylinder and using algebraic manipulation to isolate the desired variable.

The problem states that the volume of a cylinder is given by the expression $4 ackslash pi x^3$ cubic units, and its height is $x$ units. The objective is to determine which expression represents the radius of the cylinder in units. The given options are:

A. $2x$ B. $4x$ C. $2 ackslash pi x^2$ D. $4 ackslash pi x$

This problem requires us to recall the formula for the volume of a cylinder and then use algebraic techniques to solve for the radius. Let's break down the process step by step.

Understanding the Formula for the Volume of a Cylinder

The volume of a cylinder is calculated using the formula:

V = ackslash pi r^2 h

where:

  • V$ represents the **volume** of the cylinder,

  • r$ represents the **radius** of the circular base,

  • h$ represents the **height** of the cylinder, and

  • ackslash pi$ (pi) is a mathematical constant approximately equal to 3.14159.

This formula tells us that the volume is directly proportional to the square of the radius and the height. This means that if we double the radius, the volume will increase by a factor of four (since it's squared), and if we double the height, the volume will double as well. Understanding this relationship is crucial for solving problems involving cylinders.

Applying the Formula to the Given Problem

In our case, we are given the volume $V = 4 ackslash pi x^3$ and the height $h = x$. We need to find an expression for the radius $r$. To do this, we will substitute the given values into the volume formula and then solve for $r$.

Substituting the given values into the formula $V = ackslash pi r^2 h$, we get:

4 ackslash pi x^3 = ackslash pi r^2 x

Now, we need to isolate $r^2$ on one side of the equation. We can do this by dividing both sides of the equation by $ackslash pi x$:

\frac{4 ackslash pi x^3}{ackslash pi x} = \frac{ackslash pi r^2 x}{ackslash pi x}

Simplifying both sides, we get:

4x2=r24x^2 = r^2

Now, to find $r$, we need to take the square root of both sides of the equation:

4x2=r2\sqrt{4x^2} = \sqrt{r^2}

This gives us:

r=2xr = 2x

Therefore, the expression that represents the radius of the cylinder is $2x$ units.

To further clarify the solution, let's go through each step in detail:

  1. Write down the formula for the volume of a cylinder:

    V = ackslash pi r^2 h

    This formula is the foundation of our solution. It relates the volume, radius, and height of the cylinder. Knowing this formula is essential for solving any problem involving the volume of a cylinder.

  2. Substitute the given values:

    We are given that $V = 4 ackslash pi x^3$ and $h = x$. Substituting these values into the formula, we get:

    4 ackslash pi x^3 = ackslash pi r^2 x

    This step replaces the general variables in the formula with the specific values provided in the problem. This allows us to work with a concrete equation that we can solve for the unknown variable, which is the radius in this case.

  3. Divide both sides by $ackslash pi x$:

    To isolate $r^2$, we divide both sides of the equation by $ackslash pi x$:

    \frac{4 ackslash pi x^3}{ackslash pi x} = \frac{ackslash pi r^2 x}{ackslash pi x}

    This step is a crucial algebraic manipulation. Dividing both sides of an equation by the same non-zero quantity maintains the equality and helps us simplify the equation. In this case, dividing by $ackslash pi x$ cancels out those terms on the right side, bringing us closer to isolating $r^2$.

  4. Simplify the equation:

    Simplifying both sides, we get:

    4x2=r24x^2 = r^2

    Here, we perform the division and simplify the exponents. On the left side, $ackslash pi$ cancels out, and $x^3$ divided by $x$ becomes $x^2$. On the right side, the entire $ackslash pi x$ term cancels out, leaving us with just $r^2$. This simplified equation makes it easier to solve for $r$.

  5. Take the square root of both sides:

    To find $r$, we take the square root of both sides of the equation:

    4x2=r2\sqrt{4x^2} = \sqrt{r^2}

    Taking the square root is the inverse operation of squaring, and it allows us to isolate $r$. Remember that when taking the square root, we typically consider both positive and negative solutions. However, in this context, since the radius is a physical dimension, we only consider the positive solution.

  6. Solve for $r$:

    This gives us:

    r=2xr = 2x

    The square root of $4x^2$ is $2x$, and the square root of $r^2$ is $r$. Thus, we have found the expression for the radius of the cylinder in terms of $x$.

Conclusion and Answer

Therefore, the expression that represents the radius of the cylinder is $2x$ units. This corresponds to option A.

This problem demonstrates the importance of understanding and applying geometric formulas, as well as the power of algebraic manipulation in solving for unknown variables. By carefully following the steps and understanding the underlying concepts, we can successfully solve a wide range of geometry problems.

To solidify your understanding, try solving these similar problems:

  1. A cylinder has a volume of $16 ackslash pi y^3$ cubic units and a height of $4y$ units. Find the radius of the cylinder.
  2. The volume of a cylinder is $9 ackslash pi z^2$ cubic units, and the radius is $3z$ units. Determine the height of the cylinder.
  3. A cylinder has a radius of $5a$ units and a height of $2a$ units. What is the volume of the cylinder?

These problems will help you practice applying the volume of a cylinder formula and solving for different variables. Remember to break down the problem into steps, substitute the given values, and use algebraic techniques to isolate the unknown variable.

By working through these problems, you will gain confidence in your ability to solve geometry problems and further enhance your understanding of the relationships between different properties of shapes.