A Basketball Has A Surface Area Of $1810 \, \text{cm}^2$. Find Its Volume.

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Introduction

In the world of mathematics, understanding the relationship between a three-dimensional object's surface area and volume is crucial. A basketball, being a perfect sphere, presents an interesting case study. Given its surface area, we can calculate its volume using the fundamental principles of geometry. In this article, we will delve into the world of mathematics and explore the relationship between a basketball's surface area and volume.

The Surface Area of a Basketball

The surface area of a basketball is given as $1810 , \text{cm}^2$. To understand this, we need to recall the formula for the surface area of a sphere, which is $4\pi r^2$, where $r$ is the radius of the sphere. However, we are not given the radius directly. Instead, we are given the surface area, and we need to find the radius first.

Finding the Radius of the Basketball

To find the radius of the basketball, we can use the formula for the surface area of a sphere and rearrange it to solve for $r$. We have:

4Ο€r2=18104\pi r^2 = 1810

Now, we can solve for $r$:

r2=18104Ο€r^2 = \frac{1810}{4\pi}

r2=18104Γ—3.14159r^2 = \frac{1810}{4 \times 3.14159}

r2=181012.56637r^2 = \frac{1810}{12.56637}

r2=144.04r^2 = 144.04

r=144.04r = \sqrt{144.04}

r=12.00 cmr = 12.00 \, \text{cm}

The Volume of a Basketball

Now that we have the radius of the basketball, we can find its volume using the formula for the volume of a sphere, which is $\frac{4}{3}\pi r^3$. Plugging in the value of $r$, we get:

V=43Ο€(12.00)3V = \frac{4}{3}\pi (12.00)^3

V=43Ο€(1728)V = \frac{4}{3}\pi (1728)

V=43Γ—3.14159Γ—1728V = \frac{4}{3} \times 3.14159 \times 1728

V=4.18879Γ—1728V = 4.18879 \times 1728

V=7236.19 cm3V = 7236.19 \, \text{cm}^3

Conclusion

In this article, we have explored the relationship between a basketball's surface area and volume. Given the surface area of the basketball, we were able to find its radius and then use it to calculate its volume. The volume of the basketball is approximately $7236.19 , \text{cm}^3$. This demonstrates the importance of understanding the relationship between a three-dimensional object's surface area and volume in mathematics.

The Formula for the Surface Area and Volume of a Sphere

The surface area of a sphere is given by the formula:

A=4Ο€r2A = 4\pi r^2

The volume of a sphere is given by the formula:

V=43Ο€r3V = \frac{4}{3}\pi r^3

Real-World Applications

Understanding the relationship between a three-dimensional object's surface area and volume has numerous real-world applications. For example, in engineering, it is crucial to calculate the volume of a sphere to determine the amount of material needed for a project. In medicine, understanding the volume of a sphere can help in calculating the volume of organs or tumors.

Final Thoughts

In conclusion, the relationship between a basketball's surface area and volume is a fundamental concept in mathematics. By understanding this relationship, we can calculate the volume of a sphere given its surface area. This has numerous real-world applications and is an essential concept to grasp in mathematics.

References

  • [1] "Mathematics for Engineers and Scientists" by Donald R. Hill
  • [2] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [3] "Calculus: Early Transcendentals" by James Stewart

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

Introduction

In our previous article, we explored the relationship between a basketball's surface area and volume. We calculated the volume of the basketball given its surface area and discussed the importance of understanding this relationship in mathematics. In this article, we will answer some frequently asked questions related to the surface area and volume of a sphere.

Q&A

Q: What is the formula for the surface area of a sphere?

A: The formula for the surface area of a sphere is $4\pi r^2$, where $r$ is the radius of the sphere.

Q: How do I find the radius of a sphere given its surface area?

A: To find the radius of a sphere given its surface area, you can use the formula for the surface area of a sphere and rearrange it to solve for $r$. We have:

4Ο€r2=A4\pi r^2 = A

r2=A4Ο€r^2 = \frac{A}{4\pi}

r=A4Ο€r = \sqrt{\frac{A}{4\pi}}

Q: What is the formula for the volume of a sphere?

A: The formula for the volume of a sphere is $\frac{4}{3}\pi r^3$, where $r$ is the radius of the sphere.

Q: How do I find the volume of a sphere given its surface area?

A: To find the volume of a sphere given its surface area, you can first find the radius of the sphere using the formula for the surface area of a sphere, and then use the formula for the volume of a sphere. We have:

V=43Ο€r3V = \frac{4}{3}\pi r^3

Q: What are some real-world applications of understanding the relationship between a sphere's surface area and volume?

A: Understanding the relationship between a sphere's surface area and volume has numerous real-world applications. For example, in engineering, it is crucial to calculate the volume of a sphere to determine the amount of material needed for a project. In medicine, understanding the volume of a sphere can help in calculating the volume of organs or tumors.

Q: Can I use the surface area and volume formulas for any shape?

A: No, the surface area and volume formulas we discussed are specific to spheres. However, there are formulas for other shapes, such as cylinders and cones, that can be used to calculate their surface area and volume.

Q: How do I calculate the surface area and volume of a sphere with a given diameter?

A: To calculate the surface area and volume of a sphere with a given diameter, you can first find the radius of the sphere by dividing the diameter by 2, and then use the formulas for the surface area and volume of a sphere.

Conclusion

In this article, we answered some frequently asked questions related to the surface area and volume of a sphere. We discussed the formulas for the surface area and volume of a sphere, and provided examples of how to use these formulas to calculate the volume of a sphere given its surface area. We also discussed some real-world applications of understanding the relationship between a sphere's surface area and volume.

Additional Resources

  • [1] "Mathematics for Engineers and Scientists" by Donald R. Hill
  • [2] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [3] "Calculus: Early Transcendentals" by James Stewart

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

Final Thoughts

Understanding the relationship between a sphere's surface area and volume is an essential concept in mathematics. By grasping this concept, you can calculate the volume of a sphere given its surface area, and apply this knowledge to real-world problems.