The Volume Of A Cylinder Is $72\pi$ Cubic Feet, And The Radius Is 6 Feet. What Is The Height Of The Cylinder?
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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 = πr^2h, where V is the volume, π is a mathematical constant approximately equal to 3.14, r is the radius of the base of the cylinder, and h is the height of the cylinder. In this article, we will use this formula to find the height of a cylinder given its volume and radius.
The Formula for the Volume of a Cylinder
The formula for the volume of a cylinder is V = πr^2h. This formula is derived from the fact that the volume of a cylinder is equal to the area of the base times the height. The area of the base of a cylinder is given by A = πr^2, where r is the radius of the base. Therefore, the volume of a cylinder is given by V = A × h = πr^2h.
Given Information
We are given that the volume of the cylinder is 72Ï€ cubic feet and the radius is 6 feet. We can use this information to find the height of the cylinder.
Finding the Height of the Cylinder
To find the height of the cylinder, we can use the formula V = πr^2h and substitute the given values. We have V = 72π and r = 6. Substituting these values into the formula, we get:
72π = π(6)^2h
Simplifying the equation, we get:
72Ï€ = 36Ï€h
Now, we can divide both sides of the equation by 36Ï€ to solve for h:
h = 72Ï€ / 36Ï€
h = 2
Therefore, the height of the cylinder is 2 feet.
Conclusion
In this article, we used the formula for the volume of a cylinder to find the height of a cylinder given its volume and radius. We substituted the given values into the formula and solved for h. The result was that the height of the cylinder is 2 feet.
Real-World Applications
The concept of the volume of a cylinder has many real-world applications. For example, in architecture, the volume of a cylinder is used to calculate the amount of space inside a building. In engineering, the volume of a cylinder is used to calculate the amount of material needed to build a cylindrical object. In science, the volume of a cylinder is used to calculate the amount of space inside a container.
Examples
Here are a few examples of how the formula for the volume of a cylinder can be used:
- A cylindrical tank has a radius of 10 feet and a height of 5 feet. What is the volume of the tank?
- A cylindrical pipe has a radius of 2 feet and a height of 10 feet. What is the volume of the pipe?
- A cylindrical container has a radius of 5 feet and a height of 3 feet. What is the volume of the container?
Solutions
Here are the solutions to the examples:
- The volume of the tank is V = π(10)^2(5) = 1570π cubic feet.
- The volume of the pipe is V = π(2)^2(10) = 40π cubic feet.
- The volume of the container is V = π(5)^2(3) = 75π cubic feet.
Final Thoughts
In conclusion, the formula for the volume of a cylinder is a fundamental concept in mathematics that has many real-world applications. By using this formula, we can calculate the amount of space inside a cylindrical object. We can also use this formula to solve problems involving the volume of a cylinder.
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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 its volume and radius. In this article, we will answer some frequently asked questions about the volume of a cylinder.
Q&A
Q: What is the formula for the volume of a cylinder?
A: The formula for the volume of a cylinder is V = πr^2h, where V is the volume, π is a mathematical constant approximately equal to 3.14, r is the radius of the base of the cylinder, and h is the height of the cylinder.
Q: How do I calculate the volume of a cylinder?
A: To calculate the volume of a cylinder, you can use the formula V = πr^2h. Simply substitute the values of r and h into the formula and solve for V.
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 or cubic meters.
Q: Can I use the formula for the volume of a cylinder to find the radius of a cylinder?
A: Yes, you can use the formula for the volume of a cylinder to find the radius of a cylinder. If you know the volume and height of the cylinder, you can rearrange the formula to solve for r.
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 of the cylinder will also increase.
Q: Can I use the formula for the volume of a cylinder to find the height of a cylinder?
A: Yes, you can use the formula for the volume of a cylinder to find the height of a cylinder. If you know the volume and radius of the cylinder, you can rearrange the formula to solve for h.
Q: What is the significance of the constant π in the formula for the volume of a cylinder?
A: The constant π is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14 and is used in many mathematical formulas, including the formula for the volume of a cylinder.
Q: Can I use the formula for the volume of a cylinder to find the volume of a sphere?
A: No, the formula for the volume of a cylinder is not applicable to spheres. The formula for the volume of a sphere is V = (4/3)Ï€r^3, where r is the radius of the sphere.
Examples
Here are a few examples of how the formula for the volume of a cylinder can be used to answer questions:
- A cylindrical tank has a radius of 10 feet and a height of 5 feet. What is the volume of the tank?
- A cylindrical pipe has a radius of 2 feet and a height of 10 feet. What is the volume of the pipe?
- A cylindrical container has a radius of 5 feet and a height of 3 feet. What is the volume of the container?
Solutions
Here are the solutions to the examples:
- The volume of the tank is V = π(10)^2(5) = 1570π cubic feet.
- The volume of the pipe is V = π(2)^2(10) = 40π cubic feet.
- The volume of the container is V = π(5)^2(3) = 75π cubic feet.
Final Thoughts
In conclusion, the formula for the volume of a cylinder is a fundamental concept in mathematics that has many real-world applications. By using this formula, we can calculate the amount of space inside a cylindrical object. We can also use this formula to answer questions about the volume of a cylinder.