The Radius Of The Large Sphere Is Double The Radius Of The Small Sphere. How Many Times Larger Is The Volume Of The Large Sphere Compared To The Small Sphere?
Introduction
In the realm of geometry, spheres are three-dimensional objects that are perfectly round and symmetrical. The volume of a sphere is a crucial property that determines its size and capacity. In this article, we will delve into the relationship between the radii of two spheres and explore how their volumes compare.
Understanding the Problem
Let's consider two spheres: a small sphere and a large sphere. The radius of the large sphere is double the radius of the small sphere. We are asked to determine how many times larger the volume of the large sphere is compared to the small sphere.
The Formula for the Volume of a Sphere
The volume of a sphere is given by the formula:
V = (4/3)πr³
where V is the volume and r is the radius of the sphere.
Comparing the Volumes of the Two Spheres
Let's denote the radius of the small sphere as r. Then, the radius of the large sphere is 2r. We can calculate the volumes of the two spheres using the formula above:
V_small = (4/3)πr³ V_large = (4/3)π(2r)³
Simplifying the Expressions
We can simplify the expressions for the volumes of the two spheres:
V_small = (4/3)πr³ V_large = (4/3)π(8r³)
Comparing the Volumes
Now, we can compare the volumes of the two spheres by dividing the volume of the large sphere by the volume of the small sphere:
V_large / V_small = ((4/3)π(8r³)) / ((4/3)πr³)
Cancelling Out Common Factors
We can cancel out the common factors in the numerator and denominator:
V_large / V_small = 8
Conclusion
In conclusion, the volume of the large sphere is 8 times larger than the volume of the small sphere. This result makes sense, as the radius of the large sphere is double the radius of the small sphere. The formula for the volume of a sphere is a powerful tool that allows us to calculate the volumes of spheres with different radii.
Real-World Applications
The concept of comparing the volumes of spheres has numerous real-world applications. For example, in engineering, the volume of a sphere is used to calculate the capacity of containers, such as tanks and reservoirs. In physics, the volume of a sphere is used to calculate the density of objects.
Example Problems
Here are a few example problems that illustrate the concept of comparing the volumes of spheres:
- A small sphere has a radius of 5 cm. What is the volume of the sphere?
- A large sphere has a radius of 10 cm. What is the volume of the sphere?
- A small sphere has a radius of 2 cm. What is the ratio of the volume of the large sphere to the volume of the small sphere?
Solutions to Example Problems
Here are the solutions to the example problems:
- The volume of the small sphere is (4/3)π(5)³ = 523.6 cm³.
- The volume of the large sphere is (4/3)π(10)³ = 4188.8 cm³.
- The ratio of the volume of the large sphere to the volume of the small sphere is 8:1.
Conclusion
Introduction
In our previous article, we explored the relationship between the radii of two spheres and how their volumes compare. In this article, we will answer some frequently asked questions related to the topic.
Q: What is the formula for the volume of a sphere?
A: The formula for the volume of a sphere is:
V = (4/3)πr³
where V is the volume and r is the radius of the sphere.
Q: How do I calculate the volume of a sphere?
A: To calculate the volume of a sphere, you need to know its radius. You can then plug the radius into the formula:
V = (4/3)πr³
Q: What is the relationship between the radii of two spheres?
A: Let's consider two spheres: a small sphere and a large sphere. The radius of the large sphere is double the radius of the small sphere.
Q: How many times larger is the volume of the large sphere compared to the small sphere?
A: The volume of the large sphere is 8 times larger than the volume of the small sphere.
Q: What are some real-world applications of the concept of comparing the volumes of spheres?
A: The concept of comparing the volumes of spheres has numerous real-world applications. For example, in engineering, the volume of a sphere is used to calculate the capacity of containers, such as tanks and reservoirs. In physics, the volume of a sphere is used to calculate the density of objects.
Q: Can I use the formula for the volume of a sphere to calculate the volume of a cylinder?
A: No, the formula for the volume of a sphere is only applicable to spheres. To calculate the volume of a cylinder, you need to use a different formula:
V = πr²h
where V is the volume, r is the radius, and h is the height of the cylinder.
Q: What is the difference between the volume of a sphere and the volume of a cylinder?
A: The volume of a sphere is given by the formula:
V = (4/3)πr³
where V is the volume and r is the radius of the sphere.
The volume of a cylinder is given by the formula:
V = πr²h
where V is the volume, r is the radius, and h is the height of the cylinder.
Q: Can I use the formula for the volume of a sphere to calculate the volume of a cone?
A: No, the formula for the volume of a sphere is only applicable to spheres. To calculate the volume of a cone, you need to use a different formula:
V = (1/3)πr²h
where V is the volume, r is the radius, and h is the height of the cone.
Q: What is the difference between the volume of a sphere and the volume of a cone?
A: The volume of a sphere is given by the formula:
V = (4/3)πr³
where V is the volume and r is the radius of the sphere.
The volume of a cone is given by the formula:
V = (1/3)πr²h
where V is the volume, r is the radius, and h is the height of the cone.
Conclusion
In conclusion, the concept of comparing the volumes of spheres is a fundamental concept in mathematics and has numerous real-world applications. By understanding the formula for the volume of a sphere and how it relates to the radii of two spheres, we can gain a deeper understanding of the relationship between their volumes.