Solve The Formula $V=\pi R^2 H$ For $r$.A. $r=\sqrt{V H \pi}$B. $r=\sqrt{\frac{\pi H}{V}}$C. $r=\sqrt{\frac{V}{\pi H}}$D. $r=\sqrt{V-\pi H}$

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Introduction

In mathematics, solving a formula for a specific variable is an essential skill that helps us understand the relationship between different variables and make predictions or calculations. In this article, we will focus on solving the formula $V=\pi r^2 h$ for $r$. This formula is commonly used in geometry and physics to calculate the volume of a cylinder. We will explore the different options and determine the correct solution.

Understanding the Formula

The formula $V=\pi r^2 h$ represents the volume of a cylinder, where $V$ is the volume, $r$ is the radius, and $h$ is the height. To solve for $r$, we need to isolate the variable $r$ on one side of the equation.

Option A: $r=\sqrt{V h \pi}$

Let's start by examining option A: $r=\sqrt{V h \pi}$. To determine if this is the correct solution, we need to substitute the values of $V$, $h$, and $\pi$ into the equation and see if we get the correct value for $r$.

However, if we substitute the values of $V$, $h$, and $\pi$ into the equation, we get:

$r=\sqrt{V h \pi}$

$r=\sqrt{(\pi r^2 h) h}$

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

$r=\sqrt{\pi} r h$

This is not the correct solution, as it does not isolate the variable $r$.

Option B: $r=\sqrt{\frac{\pi h}{V}}$

Now, let's examine option B: $r=\sqrt{\frac{\pi h}{V}}$. To determine if this is the correct solution, we need to substitute the values of $V$, $h$, and $\pi$ into the equation and see if we get the correct value for $r$.

However, if we substitute the values of $V$, $h$, and $\pi$ into the equation, we get:

$r=\sqrt{\frac{\pi h}{V}}$

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

$r=\sqrt{\frac{\pi r^2}{V/h}}$

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Q&A: Solving the Formula for $r$

Q: What is the formula for the volume of a cylinder? A: The formula for the volume of a cylinder is $V=\pi r^2 h$, where $V$ is the volume, $r$ is the radius, and $h$ is the height.

Q: How do I solve the formula for $r$? A: To solve the formula for $r$, we need to isolate the variable $r$ on one side of the equation. We can do this by dividing both sides of the equation by $\pi r^2$.

Q: What is the correct solution for $r$? A: The correct solution for $r$ is $r=\sqrt{\frac{V}{\pi h}}$. This can be derived by dividing both sides of the equation $V=\pi r^2 h$ by $\pi h$ and then taking the square root of both sides.

Q: Why is option A incorrect? A: Option A is incorrect because it does not isolate the variable $r$ on one side of the equation. When we substitute the values of $V$, $h$, and $\pi$ into the equation, we get $r=\sqrt{V h \pi}$, which is not the correct solution.

Q: Why is option B incorrect? A: Option B is incorrect because it also does not isolate the variable $r$ on one side of the equation. When we substitute the values of $V$, $h$, and $\pi$ into the equation, we get $r=\sqrt{\frac{\pi h}{V}}$, which is not the correct solution.

Q: What is the significance of solving the formula for $r$? A: Solving the formula for $r$ is significant because it allows us to calculate the radius of a cylinder given its volume and height. This is useful in a variety of applications, such as engineering and physics.

Q: How do I apply the formula to real-world problems? A: To apply the formula to real-world problems, you need to substitute the values of $V$, $h$, and $\pi$ into the equation and solve for $r$. For example, if you know that the volume of a cylinder is 100 cubic meters and the height is 5 meters, you can use the formula to calculate the radius.

Q: What are some common mistakes to avoid when solving the formula for $r$? A: Some common mistakes to avoid when solving the formula for $r$ include:

• Not isolating the variable $r$ on one side of the equation
• Not using the correct values for $V$, $h$, and $\pi$
• Not taking the square root of both sides of the equation
• Not checking the solution for errors

Conclusion

Solving the formula for $r$ is an essential skill in mathematics and physics. By understanding the formula and how to solve it, you can calculate the radius of a cylinder given its volume and height. Remember to isolate the variable $r$ on one side of the equation and use the correct values for $V$, $h$, and $\pi$. With practice and patience, you can master this skill and apply it to real-world problems.

Additional Resources

• Mathematics formulas
• Physics formulas
• Cylinder volume calculator

Frequently Asked Questions

• Q: What is the formula for the volume of a cylinder? A: The formula for the volume of a cylinder is $V=\pi r^2 h$.
• Q: How do I solve the formula for $r$? A: To solve the formula for $r$, you need to isolate the variable $r$ on one side of the equation.
• Q: What is the correct solution for $r$? A: The correct solution for $r$ is $r=\sqrt{\frac{V}{\pi h}}$.

Related Articles

• Solving the formula for $h$
• Solving the formula for $V$
• Cylinder volume and surface area