Calculating the Volume of a Sphere with a Radius of 17

Oct 13, 2025 by talynt.org 50 views

Introduction

In mathematics, the volume of a sphere is a fundamental concept that is used to calculate the amount of space inside a sphere. The formula for the volume of a sphere is given by V = (4/3)πr^3, where r is the radius of the sphere. In this article, we will explore the different options for calculating the volume of a sphere with a radius of 17 and determine which expression gives the correct answer.

Understanding the Formula

The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius of the sphere. This formula is derived from the concept of integration and is a fundamental concept in mathematics. The formula can be broken down into three main components:

The constant (4/3) represents the proportion of the sphere's volume to its surface area.
The π (pi) represents the ratio of a circle's circumference to its diameter.
The r^3 represents the cube of the radius of the sphere.

Evaluating the Options

Now that we have a clear understanding of the formula for the volume of a sphere, let's evaluate the different options for calculating the volume of a sphere with a radius of 17.

Option A: $\frac{4}{3} \pi\left(17^2\right)$

This option is incorrect because it uses the square of the radius (17^2) instead of the cube of the radius (17^3). The correct formula for the volume of a sphere is V = (4/3)πr^3, not V = (4/3)πr^2.

Option B: $\frac{4}{3} \pi\left(17^3\right)$

This option is correct because it uses the cube of the radius (17^3) in the formula for the volume of a sphere. The correct formula for the volume of a sphere is V = (4/3)πr^3, and this option correctly implements this formula.

Option C: $4 \pi\left(17^3\right)$

This option is incorrect because it omits the constant (4/3) from the formula for the volume of a sphere. The correct formula for the volume of a sphere is V = (4/3)πr^3, not V = 4πr^3.

Option D: $4 \pi\left(17^2\right)$

This option is incorrect because it uses the square of the radius (17^2) instead of the cube of the radius (17^3) and omits the constant (4/3) from the formula for the volume of a sphere. The correct formula for the volume of a sphere is V = (4/3)πr^3, not V = 4πr^2.

Conclusion

In conclusion, the correct expression for the volume of a sphere with a radius of 17 is $\frac{4}{3} \pi\left(17^3\right)$ . This option correctly implements the formula for the volume of a sphere, which is V = (4/3)πr^3. The other options are incorrect because they either use the square of the radius instead of the cube of the radius or omit the constant (4/3) from the formula for the volume of a sphere.

Calculating the Volume of a Sphere with a Radius of 17

Now that we have determined the correct expression for the volume of a sphere with a radius of 17, let's calculate the volume of the sphere using this expression.

V = (4/3)π(17^3) V = (4/3)π(4913) V = (4/3)(3.14159)(4913) V = 20543.88

Therefore, the volume of a sphere with a radius of 17 is approximately 20543.88 cubic units.

Real-World Applications of Calculating the Volume of a Sphere

Calculating the volume of a sphere has many real-world applications, including:

Architecture: Calculating the volume of a sphere is essential in architecture to determine the amount of space inside a sphere-shaped building or structure.
Engineering: Calculating the volume of a sphere is essential in engineering to determine the amount of space inside a sphere-shaped container or vessel.
Physics: Calculating the volume of a sphere is essential in physics to determine the amount of space inside a sphere-shaped object or particle.
Mathematics: Calculating the volume of a sphere is essential in mathematics to determine the amount of space inside a sphere-shaped shape or figure.

Conclusion

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

A: The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius of the sphere.

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

A: The radius of the sphere in the problem is 17.

Q: Which expression gives the correct volume of the sphere?

A: The expression $\frac{4}{3} \pi\left(17^3\right)$ gives the correct volume of the sphere.

Q: Why is the expression $\frac{4}{3} \pi\left(17^2\right)$ incorrect?

A: The expression $\frac{4}{3} \pi\left(17^2\right)$ is incorrect because it uses the square of the radius (17^2) instead of the cube of the radius (17^3).

Q: Why is the expression $4 \pi\left(17^3\right)$ incorrect?

A: The expression $4 \pi\left(17^3\right)$ is incorrect because it omits the constant (4/3) from the formula for the volume of a sphere.

Q: Why is the expression $4 \pi\left(17^2\right)$ incorrect?

A: The expression $4 \pi\left(17^2\right)$ is incorrect because it uses the square of the radius (17^2) instead of the cube of the radius (17^3) and omits the constant (4/3) from the formula for the volume of a sphere.

Q: How do I calculate the volume of a sphere with a radius of 17?

A: To calculate the volume of a sphere with a radius of 17, you can use the expression $\frac{4}{3} \pi\left(17^3\right)$ . This expression can be evaluated as follows:

V = (4/3)π(17^3) V = (4/3)π(4913) V = (4/3)(3.14159)(4913) V = 20543.88

Therefore, the volume of a sphere with a radius of 17 is approximately 20543.88 cubic units.

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

A: Calculating the volume of a sphere has many real-world applications, including:

Architecture: Calculating the volume of a sphere is essential in architecture to determine the amount of space inside a sphere-shaped building or structure.
Engineering: Calculating the volume of a sphere is essential in engineering to determine the amount of space inside a sphere-shaped container or vessel.
Physics: Calculating the volume of a sphere is essential in physics to determine the amount of space inside a sphere-shaped object or particle.
Mathematics: Calculating the volume of a sphere is essential in mathematics to determine the amount of space inside a sphere-shaped shape or figure.

Q: Can I use a calculator to calculate the volume of a sphere?

A: Yes, you can use a calculator to calculate the volume of a sphere. Simply enter the radius of the sphere and the calculator will evaluate the expression $\frac{4}{3} \pi\left(r^3\right)$ to give you the volume of the sphere.

Q: Can I use a computer program to calculate the volume of a sphere?

A: Yes, you can use a computer program to calculate the volume of a sphere. Simply enter the radius of the sphere and the program will evaluate the expression $\frac{4}{3} \pi\left(r^3\right)$ to give you the volume of the sphere.

Conclusion

In conclusion, calculating the volume of a sphere is a fundamental concept in mathematics that has many real-world applications. The formula for the volume of a sphere is V = (4/3)πr^3, and this formula can be used to calculate the volume of a sphere with any radius. The correct expression for the volume of a sphere with a radius of 17 is $\frac{4}{3} \pi\left(17^3\right)$ , and this expression can be used to calculate the volume of the sphere using this formula.