The Radius Of A Sphere Is $\frac{1}{3}$ Ft. What Is The Approximate Volume Of The Sphere?Use $\pi \approx \frac{22}{7}$.A. $\frac{88}{567}$ Ft³ B. $\frac{88}{189}$ Ft³ C. $\frac{704}{567}$ Ft³ D. I Don't

Understanding the Problem

In this problem, we are given the radius of a sphere, which is $\frac{1}{3}$ ft. We need to find the approximate volume of the sphere using the value of $\pi \approx \frac{22}{7}$.

Recalling the Formula for the Volume of a Sphere

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

$$V = \frac{4}{3}\pi r^3$$

where $r$ is the radius of the sphere.

Substituting the Given Value of $\pi$

We are given that $\pi \approx \frac{22}{7}$. We can substitute this value into the formula for the volume of a sphere:

$$V = \frac{4}{3}\left(\frac{22}{7}\right)r^3$$

Simplifying the Expression

We can simplify the expression by multiplying the fractions:

$$V = \frac{4}{3}\left(\frac{22}{7}\right)r^3 = \frac{4 \cdot 22}{3 \cdot 7}r^3 = \frac{88}{21}r^3$$

Substituting the Given Value of $r$

We are given that the radius of the sphere is $\frac{1}{3}$ ft. We can substitute this value into the expression:

$$V = \frac{88}{21}\left(\frac{1}{3}\right)^3$$

Evaluating the Expression

We can evaluate the expression by cubing the fraction:

$$V = \frac{88}{21}\left(\frac{1}{27}\right) = \frac{88}{21 \cdot 27} = \frac{88}{567}$$

Conclusion

Therefore, the approximate volume of the sphere is $\frac{88}{567}$ ft³.

Discussion

This problem requires the application of the formula for the volume of a sphere and the substitution of the given value of $\pi$. The simplification of the expression and the evaluation of the final result are also important steps in solving this problem.

Answer

The correct answer is A. $\frac{88}{567}$ ft³.

Additional Information

The volume of a sphere is an important concept in mathematics and physics. It is used to calculate the volume of objects that are approximately spherical in shape, such as balls, spheres, and other three-dimensional objects.

Related Topics

• Volume of a Sphere: The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$.
• Approximation of $\pi$: The value of $\pi$ can be approximated as $\frac{22}{7}$.
• Simplification of Expressions: The simplification of expressions is an important step in solving mathematical problems.
• Evaluation of Final Results: The evaluation of final results is also an important step in solving mathematical problems.

References

• Mathematics Handbook: A comprehensive handbook of mathematical formulas and theorems.
• Geometry and Trigonometry: A textbook on geometry and trigonometry that covers the basics of these subjects.
• Mathematical Problems and Solutions: A collection of mathematical problems and their solutions.
  The Radius of a Sphere and Its Volume: Q&A =============================================

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

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

$$V = \frac{4}{3}\pi r^3$$

where $r$ is the radius of the sphere.

Q: What is the value of $\pi$ used in this problem?

A: The value of $\pi$ used in this problem is $\frac{22}{7}$.

Q: How do you simplify the expression for the volume of a sphere?

A: To simplify the expression for the volume of a sphere, you can multiply the fractions:

$$V = \frac{4}{3}\left(\frac{22}{7}\right)r^3 = \frac{4 \cdot 22}{3 \cdot 7}r^3 = \frac{88}{21}r^3$$

Q: What is the radius of the sphere in this problem?

A: The radius of the sphere in this problem is $\frac{1}{3}$ ft.

Q: How do you evaluate the expression for the volume of a sphere?

A: To evaluate the expression for the volume of a sphere, you can cube the fraction:

$$V = \frac{88}{21}\left(\frac{1}{27}\right) = \frac{88}{21 \cdot 27} = \frac{88}{567}$$

Q: What is the approximate volume of the sphere?

A: The approximate volume of the sphere is $\frac{88}{567}$ ft³.

Q: What is the correct answer to this problem?

A: The correct answer to this problem is A. $\frac{88}{567}$ ft³.

Q: What are some related topics to this problem?

A: Some related topics to this problem include:

• Volume of a Sphere: The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$.
• Approximation of $\pi$: The value of $\pi$ can be approximated as $\frac{22}{7}$.
• Simplification of Expressions: The simplification of expressions is an important step in solving mathematical problems.
• Evaluation of Final Results: The evaluation of final results is also an important step in solving mathematical problems.

Q: What are some references for this problem?

A: Some references for this problem include:

• Mathematics Handbook: A comprehensive handbook of mathematical formulas and theorems.
• Geometry and Trigonometry: A textbook on geometry and trigonometry that covers the basics of these subjects.
• Mathematical Problems and Solutions: A collection of mathematical problems and their solutions.

Q: What is the importance of the volume of a sphere in mathematics and physics?

A: The volume of a sphere is an important concept in mathematics and physics. It is used to calculate the volume of objects that are approximately spherical in shape, such as balls, spheres, and other three-dimensional objects.

Q: How do you calculate the volume of a sphere in real-world applications?

A: To calculate the volume of a sphere in real-world applications, you can use the formula $V = \frac{4}{3}\pi r^3$ and substitute the given value of the radius. You can also use a calculator or a computer program to evaluate the expression.

Q: What are some common mistakes to avoid when calculating the volume of a sphere?

A: Some common mistakes to avoid when calculating the volume of a sphere include:

• Rounding errors: Rounding errors can occur when evaluating the expression for the volume of a sphere.
• Incorrect values of $\pi$: Using an incorrect value of $\pi$ can lead to incorrect results.
• Incorrect values of the radius: Using an incorrect value of the radius can lead to incorrect results.

Q: How do you verify the accuracy of the results when calculating the volume of a sphere?

A: To verify the accuracy of the results when calculating the volume of a sphere, you can:

• Check the units: Check that the units of the result are correct.
• Check the magnitude: Check that the magnitude of the result is reasonable.
• Check the precision: Check that the precision of the result is sufficient.

Q: What are some real-world applications of the volume of a sphere?

A: Some real-world applications of the volume of a sphere include:

• Designing containers: The volume of a sphere is used to design containers that are approximately spherical in shape.
• Calculating the volume of objects: The volume of a sphere is used to calculate the volume of objects that are approximately spherical in shape.
• Solving problems in physics: The volume of a sphere is used to solve problems in physics, such as calculating the volume of a sphere that is rotating at a certain speed.