The Volume Of A Cylinder Is 126 Π Ft 3 126 \pi \, \text{ft}^3 126 Π Ft 3 And The Radius Of The Circular Base Is 6 Ft. What Is The Height Of The Cylinder?$[ \begin{array}{l} V = B \cdot H \ 126 \pi = (6)^2 \cdot \pi \cdot H \ 126 \pi = 36 \pi \cdot H \ 126
Introduction
In mathematics, the volume of a cylinder is a fundamental concept that is used to calculate the amount of space inside a cylindrical object. The formula for the volume of a cylinder is given by V = B * h, where V is the volume, B is the area of the circular base, and h is the height of the cylinder. In this article, we will use the given volume of a cylinder and the radius of the circular base to find the height of the cylinder.
Understanding the Formula
The formula for the volume of a cylinder is V = B * h, where B is the area of the circular base. The area of a circle is given by A = π * r^2, where r is the radius of the circle. Therefore, the formula for the volume of a cylinder can be rewritten as V = π * r^2 * h.
Given Information
The volume of the cylinder is given as 126π ft^3, and the radius of the circular base is given as 6 ft. We can use this information to find the height of the cylinder.
Calculating the Height
Using the formula V = π * r^2 * h, we can substitute the given values to get:
126π = π * (6)^2 * h
Simplifying the equation, we get:
126π = 36π * h
To find the height, we can divide both sides of the equation by 36π:
h = 126π / 36π
h = 3.5
Therefore, the height of the cylinder is 3.5 ft.
Conclusion
In this article, we used the formula for the volume of a cylinder to find the height of the cylinder given the volume and the radius of the circular base. We simplified the equation and solved for the height, which came out to be 3.5 ft. This problem demonstrates the importance of understanding the formula for the volume of a cylinder and how to use it to solve real-world problems.
Real-World Applications
The formula for the volume of a cylinder has many real-world applications, such as:
- Calculating the volume of a water tank
- Finding the volume of a cylindrical container
- Determining the volume of a cylindrical pipe
- Calculating the volume of a cylindrical building
Tips and Tricks
When solving problems involving the volume of a cylinder, make sure to:
- Use the correct formula: V = π * r^2 * h
- Substitute the given values correctly
- Simplify the equation to solve for the height
- Check your units to ensure they are consistent
Practice Problems
Try solving the following problems to practice your skills:
- Find the height of a cylinder with a volume of 200π ft^3 and a radius of 8 ft.
- Calculate the volume of a cylinder with a height of 4 ft and a radius of 2 ft.
- Find the height of a cylinder with a volume of 300π ft^3 and a radius of 10 ft.
References
- "Mathematics for Engineers and Scientists" by Donald R. Hill
- "Calculus" by Michael Spivak
- "Geometry and Trigonometry" by I.M. Gelfand
Discussion
Introduction
In our previous article, we discussed the formula for the volume of a cylinder and used it to find the height of a cylinder given the volume and the radius of the circular base. In this article, we will answer some frequently asked questions about the volume of a cylinder.
Q: What is the formula for the volume of a cylinder?
A: The formula for the volume of a cylinder is V = π * r^2 * h, where V is the volume, π is a mathematical constant approximately equal to 3.14, r is the radius of the circular base, and h is the height of the cylinder.
Q: What is the relationship between the volume of a cylinder and its height?
A: The volume of a cylinder is directly proportional to its height. This means that if the height of the cylinder increases, the volume will also increase.
Q: How do I calculate the volume of a cylinder if I know the radius and the height?
A: To calculate the volume of a cylinder, you can use the formula V = π * r^2 * h. Simply substitute the values of the radius and the height into the formula and solve for the volume.
Q: What is the unit of measurement for the volume of a cylinder?
A: The unit of measurement for the volume of a cylinder is typically cubic units, such as cubic feet (ft^3) or cubic meters (m^3).
Q: Can I use the formula for the volume of a cylinder to find the radius of the circular base?
A: Yes, you can use the formula for the volume of a cylinder to find the radius of the circular base. However, you will need to know the volume and the height of the cylinder. You can rearrange the formula to solve for the radius: r = √(V / (π * h)).
Q: What are some real-world applications of the formula for the volume of a cylinder?
A: The formula for the volume of a cylinder has many real-world applications, such as:
- Calculating the volume of a water tank
- Finding the volume of a cylindrical container
- Determining the volume of a cylindrical pipe
- Calculating the volume of a cylindrical building
Q: How do I use the formula for the volume of a cylinder to solve problems involving cylindrical objects?
A: To use the formula for the volume of a cylinder to solve problems involving cylindrical objects, follow these steps:
- Identify the given information, such as the volume, radius, and height of the cylinder.
- Use the formula V = π * r^2 * h to calculate the volume of the cylinder.
- Rearrange the formula to solve for the unknown variable, such as the radius or the height.
- Substitute the values into the formula and solve for the unknown variable.
Q: What are some common mistakes to avoid when using the formula for the volume of a cylinder?
A: Some common mistakes to avoid when using the formula for the volume of a cylinder include:
- Using the wrong formula: Make sure to use the correct formula, V = π * r^2 * h.
- Substituting the wrong values: Double-check that you are using the correct values for the radius and the height.
- Failing to simplify the equation: Make sure to simplify the equation to solve for the unknown variable.
Conclusion
In this article, we answered some frequently asked questions about the volume of a cylinder. We discussed the formula for the volume of a cylinder, its relationship to the height of the cylinder, and some common mistakes to avoid when using the formula. We also provided some real-world applications of the formula and tips for using it to solve problems involving cylindrical objects.
Practice Problems
Try solving the following problems to practice your skills:
- Find the volume of a cylinder with a radius of 4 ft and a height of 6 ft.
- Calculate the radius of a cylinder with a volume of 200π ft^3 and a height of 8 ft.
- Find the height of a cylinder with a volume of 300π ft^3 and a radius of 10 ft.
References
- "Mathematics for Engineers and Scientists" by Donald R. Hill
- "Calculus" by Michael Spivak
- "Geometry and Trigonometry" by I.M. Gelfand
Discussion
What are some other questions you have about the volume of a cylinder? Share your thoughts and experiences in the comments below.