The Volume Of A Sphere Is Given By $\frac{4}{3} \pi R^3$, Where $r$ Is The Radius.The Bouncy Ball Below Is A Sphere With A Volume Of $4200 \pi \text{ Mm}^3$.Calculate Its Radius, $r$. If Your Answer Is A Decimal,

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Introduction

In mathematics, the volume of a sphere is a fundamental concept that has numerous applications in various fields, including physics, engineering, and computer science. The formula for the volume of a sphere is given by 43πr3\frac{4}{3} \pi r^3, where rr is the radius of the sphere. In this article, we will discuss how to calculate the radius of a sphere given its volume.

The Formula for the Volume of a Sphere

The formula for the volume of a sphere is given by:

V=43πr3V = \frac{4}{3} \pi r^3

where VV is the volume of the sphere and rr is the radius.

Calculating the Radius of a Sphere

To calculate the radius of a sphere, we can rearrange the formula for the volume of a sphere to solve for rr. We can do this by dividing both sides of the equation by 43π\frac{4}{3} \pi:

r3=3V4πr^3 = \frac{3V}{4\pi}

Taking the cube root of both sides of the equation, we get:

r=3V4π3r = \sqrt[3]{\frac{3V}{4\pi}}

Calculating the Radius of a Bouncy Ball

Now, let's apply this formula to calculate the radius of a bouncy ball with a volume of 4200π mm34200 \pi \text{ mm}^3. Plugging in the value of VV into the formula, we get:

r=3(4200π)4π3r = \sqrt[3]{\frac{3(4200 \pi)}{4\pi}}

Simplifying the expression, we get:

r=1260043r = \sqrt[3]{\frac{12600}{4}}

r=31503r = \sqrt[3]{3150}

r≈15.6 mmr \approx 15.6 \text{ mm}

Conclusion

In this article, we discussed how to calculate the radius of a sphere given its volume. We used the formula for the volume of a sphere to derive a formula for the radius of a sphere. We then applied this formula to calculate the radius of a bouncy ball with a volume of 4200π mm34200 \pi \text{ mm}^3. The result was a radius of approximately 15.6 mm15.6 \text{ mm}.

The Importance of Calculating the Radius of a Sphere

Calculating the radius of a sphere is an important concept in mathematics and has numerous applications in various fields. For example, in physics, the radius of a sphere can be used to calculate the volume of a planet or a star. In engineering, the radius of a sphere can be used to design and optimize the shape of a container or a machine part. In computer science, the radius of a sphere can be used to calculate the distance between two points in 3D space.

Real-World Applications of Calculating the Radius of a Sphere

Calculating the radius of a sphere has numerous real-world applications. For example:

  • Designing Containers: Calculating the radius of a sphere can be used to design and optimize the shape of a container or a machine part.
  • Physics and Engineering: Calculating the radius of a sphere can be used to calculate the volume of a planet or a star.
  • Computer Science: Calculating the radius of a sphere can be used to calculate the distance between two points in 3D space.
  • Medical Imaging: Calculating the radius of a sphere can be used to analyze medical images and diagnose diseases.

Conclusion

In conclusion, calculating the radius of a sphere is an important concept in mathematics that has numerous applications in various fields. We used the formula for the volume of a sphere to derive a formula for the radius of a sphere and then applied this formula to calculate the radius of a bouncy ball with a volume of 4200π mm34200 \pi \text{ mm}^3. The result was a radius of approximately 15.6 mm15.6 \text{ mm}. We also discussed the importance of calculating the radius of a sphere and its numerous real-world applications.

References

  • Mathematics: The volume of a sphere is a fundamental concept in mathematics that has numerous applications in various fields.
  • Physics: Calculating the radius of a sphere can be used to calculate the volume of a planet or a star.
  • Engineering: Calculating the radius of a sphere can be used to design and optimize the shape of a container or a machine part.
  • Computer Science: Calculating the radius of a sphere can be used to calculate the distance between two points in 3D space.

Frequently Asked Questions

  • What is the formula for the volume of a sphere?
    • The formula for the volume of a sphere is given by 43πr3\frac{4}{3} \pi r^3, where rr is the radius of the sphere.
  • How do I calculate the radius of a sphere?
    • To calculate the radius of a sphere, we can rearrange the formula for the volume of a sphere to solve for rr. We can do this by dividing both sides of the equation by 43π\frac{4}{3} \pi.
  • What are the real-world applications of calculating the radius of a sphere?
    • Calculating the radius of a sphere has numerous real-world applications, including designing containers, physics and engineering, computer science, and medical imaging.