A Sphere Has A Diameter Of 12 Ft. What Is The Volume Of The Sphere? Give The Exact Value In Terms Of Π \pi Π .Express Your Answer In Cubic Feet.
Introduction
The volume of a sphere is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and architecture. In this article, we will explore the formula for calculating the volume of a sphere and apply it to a specific problem. We will also discuss the importance of the diameter of a sphere in determining its volume.
Formula for the Volume of a Sphere
The formula for the volume of a sphere is given by:
V = (4/3)πr^3
where V is the volume of the sphere, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.
Relationship Between Diameter and Radius
The diameter of a sphere is twice the radius, and it is denoted by the symbol d. Therefore, we can write:
d = 2r
Given Information
We are given that the diameter of the sphere is 12 ft. We can use this information to find the radius of the sphere.
Finding the Radius of the Sphere
Using the relationship between diameter and radius, we can write:
d = 2r 12 = 2r r = 12/2 r = 6
Calculating the Volume of the Sphere
Now that we have found the radius of the sphere, we can use the formula for the volume of a sphere to calculate its volume.
V = (4/3)πr^3 V = (4/3)π(6)^3 V = (4/3)π(216) V = (4/3)(3.14)(216) V = 904.7784
Expressing the Answer in Terms of π
We are asked to express our answer in terms of π. Therefore, we can write:
V = (4/3)π(216) V = (4/3)(216)π V = 288π
Conclusion
In this article, we have explored the formula for calculating the volume of a sphere and applied it to a specific problem. We have also discussed the importance of the diameter of a sphere in determining its volume. We have found that the volume of the sphere is 288π cubic feet.
Importance of the Diameter of a Sphere
The diameter of a sphere is a critical parameter in determining its volume. A larger diameter results in a larger volume, and vice versa. This is because the diameter is directly proportional to the radius, and the radius is the key factor in determining the volume of a sphere.
Applications of the Volume of a Sphere
The volume of a sphere has numerous applications in various fields such as physics, engineering, and architecture. For example, in physics, the volume of a sphere is used to calculate the mass of a planet or a star. In engineering, the volume of a sphere is used to design containers and tanks. In architecture, the volume of a sphere is used to design domes and spheres.
Real-World Examples
The volume of a sphere has numerous real-world applications. For example, a water tank with a diameter of 12 ft has a volume of 288π cubic feet. A sphere with a diameter of 6 ft has a volume of 288π/8 cubic feet. A sphere with a diameter of 18 ft has a volume of 288π(3/2) cubic feet.
Limitations of the Formula
The formula for the volume of a sphere is an approximation, and it assumes that the sphere is a perfect sphere. In reality, spheres are not perfect, and they have imperfections such as bumps and scratches. These imperfections can affect the volume of the sphere, and they must be taken into account when calculating the volume of a sphere.
Conclusion
In conclusion, the volume of a sphere is a critical parameter in determining its size and shape. The formula for the volume of a sphere is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and architecture. We have explored the formula for calculating the volume of a sphere and applied it to a specific problem. We have also discussed the importance of the diameter of a sphere in determining its volume.
Introduction
In our previous article, we explored the formula for calculating the volume of a sphere and applied it to a specific problem. We also discussed the importance of the diameter of a sphere in determining its volume. In this article, we will answer some frequently asked questions about the volume of a sphere.
Q: What is the formula for the volume of a sphere?
A: The formula for the volume of a sphere is given by:
V = (4/3)πr^3
where V is the volume of the sphere, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.
Q: What is the relationship between the diameter and radius of a sphere?
A: The diameter of a sphere is twice the radius, and it is denoted by the symbol d. Therefore, we can write:
d = 2r
Q: How do I find the radius of a sphere if I know its diameter?
A: To find the radius of a sphere, you can use the relationship between the diameter and radius:
d = 2r r = d/2
Q: What is the volume of a sphere with a diameter of 12 ft?
A: To find the volume of a sphere with a diameter of 12 ft, you can use the formula for the volume of a sphere:
V = (4/3)πr^3 V = (4/3)π(6)^3 V = (4/3)π(216) V = (4/3)(3.14)(216) V = 904.7784
Q: Can I express the answer in terms of π?
A: Yes, you can express the answer in terms of π. Therefore, we can write:
V = (4/3)π(216) V = (4/3)(216)π V = 288π
Q: What is the importance of the diameter of a sphere in determining its volume?
A: The diameter of a sphere is a critical parameter in determining its volume. A larger diameter results in a larger volume, and vice versa. This is because the diameter is directly proportional to the radius, and the radius is the key factor in determining the volume of a sphere.
Q: What are some real-world applications of the volume of a sphere?
A: The volume of a sphere has numerous real-world applications. For example, in physics, the volume of a sphere is used to calculate the mass of a planet or a star. In engineering, the volume of a sphere is used to design containers and tanks. In architecture, the volume of a sphere is used to design domes and spheres.
Q: What are some limitations of the formula for the volume of a sphere?
A: The formula for the volume of a sphere is an approximation, and it assumes that the sphere is a perfect sphere. In reality, spheres are not perfect, and they have imperfections such as bumps and scratches. These imperfections can affect the volume of the sphere, and they must be taken into account when calculating the volume of a sphere.
Q: How do I calculate the volume of a sphere with a diameter of 6 ft?
A: To calculate the volume of a sphere with a diameter of 6 ft, you can use the formula for the volume of a sphere:
V = (4/3)πr^3 V = (4/3)π(3)^3 V = (4/3)π(27) V = (4/3)(3.14)(27) V = 113.0973
Q: How do I calculate the volume of a sphere with a diameter of 18 ft?
A: To calculate the volume of a sphere with a diameter of 18 ft, you can use the formula for the volume of a sphere:
V = (4/3)πr^3 V = (4/3)π(9)^3 V = (4/3)π(729) V = (4/3)(3.14)(729) V = 3054.0832
Conclusion
In conclusion, the volume of a sphere is a critical parameter in determining its size and shape. The formula for the volume of a sphere is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and architecture. We have answered some frequently asked questions about the volume of a sphere and provided examples of how to calculate the volume of a sphere with different diameters.