The Volume Of A Cylinder Can Be Found Using The Formula $V = \pi R^2 H$.Which Is The Correct First Step In Finding The Height Of A Cylinder With A Volume Of $192 \pi$ Cubic Inches And A Radius Of 5 Inches?A. $5 = \pi(192)^2

Introduction

The volume of a cylinder is a fundamental concept in mathematics, and it is essential to understand how to calculate it. The formula for the volume of a cylinder is given by $V = \pi r^2 h$, where $V$ is the volume, $r$ is the radius, and $h$ is the height. In this article, we will focus on finding the height of a cylinder with a given volume and radius.

Understanding the Formula

The formula for the volume of a cylinder is $V = \pi r^2 h$. To find the height of the cylinder, we need to rearrange this formula to isolate $h$. We can do this by dividing both sides of the equation by $\pi r^2$, which gives us:

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

Finding the Height of a Cylinder

Now that we have the formula for the height of a cylinder, we can use it to find the height of a cylinder with a given volume and radius. Let's consider an example where the volume of the cylinder is $192 \pi$ cubic inches and the radius is 5 inches.

Step 1: Plug in the Values

To find the height of the cylinder, we need to plug in the values of $V$ and $r$ into the formula. We have $V = 192 \pi$ and $r = 5$. Plugging these values into the formula, we get:

$$h = \frac{192 \pi}{\pi (5)^2}$$

Step 2: Simplify the Expression

Now that we have plugged in the values, we can simplify the expression. We can start by canceling out the $\pi$ terms, which gives us:

$$h = \frac{192}{(5)^2}$$

Step 3: Evaluate the Expression

Now that we have simplified the expression, we can evaluate it. We can start by calculating the value of $(5)^2$, which is 25. Then, we can divide 192 by 25, which gives us:

$$h = \frac{192}{25} = 7.68$$

Conclusion

In this article, we have shown how to find the height of a cylinder with a given volume and radius. We started by understanding the formula for the volume of a cylinder, and then we rearranged it to isolate the height. We then used this formula to find the height of a cylinder with a volume of $192 \pi$ cubic inches and a radius of 5 inches. The height of the cylinder is approximately 7.68 inches.

Discussion

What do you think about the formula for the volume of a cylinder? Do you have any questions or comments about finding the height of a cylinder? Let us know in the discussion section below.

Discussion Section

• Question 1: What is the formula for the volume of a cylinder?
• Answer 1: The formula for the volume of a cylinder is $V = \pi r^2 h$.
• Question 2: How do you find the height of a cylinder?
• Answer 2: To find the height of a cylinder, you need to rearrange the formula for the volume of a cylinder to isolate $h$. You can do this by dividing both sides of the equation by $\pi r^2$.
• Question 3: What is the height of a cylinder with a volume of $192 \pi$ cubic inches and a radius of 5 inches?
• Answer 3: The height of the cylinder is approximately 7.68 inches.

Related Topics

• Volume of a Sphere: The volume of a sphere is given by the formula $V = \frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere.
• Surface Area of a Cylinder: The surface area of a cylinder is given by the formula $A = 2 \pi r^2 + 2 \pi rh$, where $r$ is the radius and $h$ is the height of the cylinder.
• Volume of a Cone: The volume of a cone is given by the formula $V = \frac{1}{3} \pi r^2 h$, where $r$ is the radius and $h$ is the height of the cone.
  The Volume of a Cylinder: A Q&A Guide =============================================

Introduction

In our previous article, we discussed how to find the height of a cylinder with a given volume and radius. In this article, we will provide a Q&A guide to help you understand the concept of the volume of a cylinder and how to calculate it.

Q&A Guide

Q1: What is the formula for the volume of a cylinder?

A1: 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.

Q2: How do you find the height of a cylinder?

A2: To find the height of a cylinder, you need to rearrange the formula for the volume of a cylinder to isolate $h$. You can do this by dividing both sides of the equation by $\pi r^2$.

Q3: What is the height of a cylinder with a volume of $192 \pi$ cubic inches and a radius of 5 inches?

A3: The height of the cylinder is approximately 7.68 inches.

Q4: What is the volume of a cylinder with a radius of 4 inches and a height of 6 inches?

A4: To find the volume of the cylinder, we need to plug in the values of $r$ and $h$ into the formula. We have $r = 4$ and $h = 6$. Plugging these values into the formula, we get:

$$V = \pi (4)^2 (6)$$

Simplifying the expression, we get:

$$V = \pi (16) (6)$$

$$V = 96 \pi$$

Q5: What is the surface area of a cylinder with a radius of 3 inches and a height of 8 inches?

A5: To find the surface area of the cylinder, we need to use the formula $A = 2 \pi r^2 + 2 \pi rh$. We have $r = 3$ and $h = 8$. Plugging these values into the formula, we get:

$$A = 2 \pi (3)^2 + 2 \pi (3) (8)$$

Simplifying the expression, we get:

$$A = 2 \pi (9) + 2 \pi (24)$$

$$A = 18 \pi + 48 \pi$$

$$A = 66 \pi$$

Q6: What is the volume of a cylinder with a radius of 2 inches and a height of 10 inches?

A6: To find the volume of the cylinder, we need to plug in the values of $r$ and $h$ into the formula. We have $r = 2$ and $h = 10$. Plugging these values into the formula, we get:

$$V = \pi (2)^2 (10)$$

Simplifying the expression, we get:

$$V = \pi (4) (10)$$

$$V = 40 \pi$$

Q7: What is the surface area of a cylinder with a radius of 5 inches and a height of 12 inches?

A7: To find the surface area of the cylinder, we need to use the formula $A = 2 \pi r^2 + 2 \pi rh$. We have $r = 5$ and $h = 12$. Plugging these values into the formula, we get:

$$A = 2 \pi (5)^2 + 2 \pi (5) (12)$$

Simplifying the expression, we get:

$$A = 2 \pi (25) + 2 \pi (60)$$

$$A = 50 \pi + 120 \pi$$

$$A = 170 \pi$$

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of the volume of a cylinder and how to calculate it. We have answered questions about the formula for the volume of a cylinder, how to find the height of a cylinder, and how to calculate the surface area of a cylinder.

Related Topics

• Volume of a Sphere: The volume of a sphere is given by the formula $V = \frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere.
• Surface Area of a Cylinder: The surface area of a cylinder is given by the formula $A = 2 \pi r^2 + 2 \pi rh$, where $r$ is the radius and $h$ is the height of the cylinder.
• Volume of a Cone: The volume of a cone is given by the formula $V = \frac{1}{3} \pi r^2 h$, where $r$ is the radius and $h$ is the height of the cone.

Discussion

What do you think about the formula for the volume of a cylinder? Do you have any questions or comments about finding the height of a cylinder or calculating the surface area of a cylinder? Let us know in the discussion section below.

Discussion Section

• Question 1: What is the formula for the volume of a cylinder?
• Answer 1: The formula for the volume of a cylinder is $V = \pi r^2 h$.
• Question 2: How do you find the height of a cylinder?
• Answer 2: To find the height of a cylinder, you need to rearrange the formula for the volume of a cylinder to isolate $h$. You can do this by dividing both sides of the equation by $\pi r^2$.
• Question 3: What is the height of a cylinder with a volume of $192 \pi$ cubic inches and a radius of 5 inches?
• Answer 3: The height of the cylinder is approximately 7.68 inches.