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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 architecture. The volume of a sphere is given by the formula $v(r) = \frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere. In this article, we will explore what $v\left(\frac{5}{7}\right)$ represents.

The Formula for the Volume of a Sphere

The formula for the volume of a sphere is given by $v(r) = \frac{4}{3} \pi r^3$. This formula is derived from the fact that the volume of a sphere is proportional to the cube of its radius. The constant of proportionality is $\frac{4}{3} \pi$.

What does $v\left(\frac{5}{7}\right)$ Represent?

To understand what $v\left(\frac{5}{7}\right)$ represents, we need to substitute $\frac{5}{7}$ into the formula for the volume of a sphere. This gives us:

$v\left(\frac{5}{7}\right) = \frac{4}{3} \pi \left(\frac{5}{7}\right)^3$

Evaluating the Expression

To evaluate the expression, we need to calculate the cube of $\frac{5}{7}$ and then multiply it by $\frac{4}{3} \pi$. This gives us:

$v\left(\frac{5}{7}\right) = \frac{4}{3} \pi \left(\frac{125}{343}\right)$

Simplifying the Expression

To simplify the expression, we can multiply the numerator and denominator by $343$ to get rid of the fraction in the denominator. This gives us:

$v\left(\frac{5}{7}\right) = \frac{4}{3} \pi \left(\frac{125 \times 343}{343 \times 343}\right)$

$v\left(\frac{5}{7}\right) = \frac{4}{3} \pi \left(\frac{43025}{117649}\right)$

The Final Answer

Therefore, $v\left(\frac{5}{7}\right)$ represents the volume of a sphere with a radius of $\frac{5}{7}$.

Conclusion

In conclusion, $v\left(\frac{5}{7}\right)$ represents the volume of a sphere with a radius of $\frac{5}{7}$. This is calculated using the formula for the volume of a sphere, which is $v(r) = \frac{4}{3} \pi r^3$. The expression is evaluated by substituting $\frac{5}{7}$ into the formula and simplifying the resulting expression.

The Importance of Understanding the Volume of a Sphere

Understanding the volume of a sphere is crucial in various fields, including physics, engineering, and architecture. The volume of a sphere is used to calculate the amount of material required to build a sphere, as well as the amount of space occupied by a sphere. In addition, the volume of a sphere is used to calculate the pressure exerted by a sphere on its surroundings.

Real-World Applications of the Volume of a Sphere

The volume of a sphere has numerous real-world applications. For example, in architecture, the volume of a sphere is used to calculate the amount of material required to build a dome or a sphere-shaped building. In engineering, the volume of a sphere is used to calculate the amount of space occupied by a sphere-shaped tank or a sphere-shaped container. In physics, the volume of a sphere is used to calculate the pressure exerted by a sphere on its surroundings.

Conclusion

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Introduction

In our previous article, we explored the concept of the volume of a sphere and how it can be calculated using the formula $v(r) = \frac{4}{3} \pi r^3$. In this article, we will answer some frequently asked questions about the volume of a sphere.

Q: What is the volume of a sphere with a radius of 5 cm?

A: To calculate the volume of a sphere with a radius of 5 cm, we can use the formula $v(r) = \frac{4}{3} \pi r^3$. Substituting $r = 5$ into the formula, we get:

$v(5) = \frac{4}{3} \pi (5)^3$

$v(5) = \frac{4}{3} \pi (125)$

$v(5) = \frac{500}{3} \pi$

$v(5) \approx 523.6$ cubic centimeters

Q: What is the volume of a sphere with a radius of 10 mm?

A: To calculate the volume of a sphere with a radius of 10 mm, we can use the formula $v(r) = \frac{4}{3} \pi r^3$. Substituting $r = 10$ into the formula, we get:

$v(10) = \frac{4}{3} \pi (10)^3$

$v(10) = \frac{4}{3} \pi (1000)$

$v(10) = \frac{4000}{3} \pi$

$v(10) \approx 4188.8$ cubic millimeters

Q: How does the volume of a sphere change when the radius is doubled?

A: To understand how the volume of a sphere changes when the radius is doubled, we can use the formula $v(r) = \frac{4}{3} \pi r^3$. If we double the radius, the new radius is $2r$. Substituting $2r$ into the formula, we get:

$v(2r) = \frac{4}{3} \pi (2r)^3$

$v(2r) = \frac{4}{3} \pi (8r^3)$

$v(2r) = 8 \times \frac{4}{3} \pi r^3$

$v(2r) = 8v(r)$

This means that when the radius of a sphere is doubled, its volume increases by a factor of 8.

Q: What is the volume of a sphere with a radius of 0.5 cm?

A: To calculate the volume of a sphere with a radius of 0.5 cm, we can use the formula $v(r) = \frac{4}{3} \pi r^3$. Substituting $r = 0.5$ into the formula, we get:

$v(0.5) = \frac{4}{3} \pi (0.5)^3$

$v(0.5) = \frac{4}{3} \pi (0.125)$

$v(0.5) = \frac{0.5}{3} \pi$

$v(0.5) \approx 0.5236$ cubic centimeters

Q: How does the volume of a sphere change when the radius is halved?

A: To understand how the volume of a sphere changes when the radius is halved, we can use the formula $v(r) = \frac{4}{3} \pi r^3$. If we halve the radius, the new radius is $\frac{r}{2}$. Substituting $\frac{r}{2}$ into the formula, we get:

$v\left(\frac{r}{2}\right) = \frac{4}{3} \pi \left(\frac{r}{2}\right)^3$

$v\left(\frac{r}{2}\right) = \frac{4}{3} \pi \left(\frac{r^3}{8}\right)$

$v\left(\frac{r}{2}\right) = \frac{1}{8} \times \frac{4}{3} \pi r^3$

$v\left(\frac{r}{2}\right) = \frac{1}{8}v(r)$

This means that when the radius of a sphere is halved, its volume decreases by a factor of 8.

Conclusion

In conclusion, the volume of a sphere is a fundamental concept that has numerous applications in various fields. The formula for the volume of a sphere is $v(r) = \frac{4}{3} \pi r^3$. We have answered some frequently asked questions about the volume of a sphere, including how the volume changes when the radius is doubled or halved.