What Is The Volume Of A Sphere With A Surface Area Of $2500 \pi \, \text{cm}^2$?A. $\frac{208331}{3} \pi \, \text{cm}^3$B. $2500 \pi \, \text{cm}^3$C. $\frac{20862}{3} \pi \, \text{cm}^3$D. $\frac{166662}{3}

Introduction

In mathematics, the surface area and volume of a sphere are two fundamental concepts that are often used to describe the properties of a sphere. The surface area of a sphere is the total area of its surface, while the volume of a sphere is the amount of space inside the sphere. In this article, we will explore the relationship between the surface area and volume of a sphere and use this relationship to find the volume of a sphere with a given surface area.

Surface Area of a Sphere

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

$$A = 4 \pi r^2$$

where $A$ is the surface area and $r$ is the radius of the sphere.

Volume of a Sphere

The volume of a sphere is given by the formula:

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

where $V$ is the volume and $r$ is the radius of the sphere.

Given Surface Area

We are given that the surface area of the sphere is $2500 \pi \, \text{cm}^2$. We can use this information to find the radius of the sphere.

Finding the Radius

We can set up an equation using the formula for the surface area of a sphere:

$$4 \pi r^2 = 2500 \pi$$

We can simplify this equation by dividing both sides by $4 \pi$:

$$r^2 = 625$$

Taking the square root of both sides, we get:

$$r = 25$$

Finding the Volume

Now that we have found the radius of the sphere, we can use the formula for the volume of a sphere to find the volume:

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

Substituting $r = 25$, we get:

$$V = \frac{4}{3} \pi (25)^3$$

Simplifying this expression, we get:

$$V = \frac{4}{3} \pi (15625)$$

$$V = \frac{62500}{3} \pi$$

$$V = \frac{208331}{3} \pi \, \text{cm}^3$$

Conclusion

In this article, we used the relationship between the surface area and volume of a sphere to find the volume of a sphere with a given surface area. We found that the volume of the sphere is $\frac{208331}{3} \pi \, \text{cm}^3$. This is the correct answer, and it is option A.

Comparison of Options

Let's compare our answer with the other options:

• Option A: $\frac{208331}{3} \pi \, \text{cm}^3$
• Option B: $2500 \pi \, \text{cm}^3$
• Option C: $\frac{20862}{3} \pi \, \text{cm}^3$
• Option D: $\frac{166662}{3} \pi \, \text{cm}^3$

Our answer is option A, which is the correct answer.

Final Answer

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Introduction

In our previous article, we explored the relationship between the surface area and volume of a sphere and used this relationship to find the volume of a sphere with a given surface area. In this article, we will answer some frequently asked questions related to the volume of a sphere with a given surface area.

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

A: The formula for the surface area of a sphere is:

$$A = 4 \pi r^2$$

where $A$ is the surface area and $r$ is the radius of the sphere.

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

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

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

where $V$ is the volume and $r$ is the radius of the sphere.

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

A: To find the radius of a sphere with a given surface area, you can use the formula for the surface area of a sphere:

$$4 \pi r^2 = A$$

where $A$ is the surface area. You can then solve for $r$.

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

A: To find the volume of a sphere with a given surface area, you can first find the radius of the sphere using the formula for the surface area of a sphere. Then, you can use the formula for the volume of a sphere:

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

Q: What is the relationship between the surface area and volume of a sphere?

A: The surface area and volume of a sphere are related by the formulas:

$$A = 4 \pi r^2$$

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

These formulas show that the surface area and volume of a sphere are both functions of the radius of the sphere.

Q: How do I calculate the volume of a sphere with a given surface area in terms of pi?

A: To calculate the volume of a sphere with a given surface area in terms of pi, you can first find the radius of the sphere using the formula for the surface area of a sphere. Then, you can use the formula for the volume of a sphere:

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

For example, if the surface area of the sphere is $2500 \pi \, \text{cm}^2$, you can find the radius of the sphere and then use the formula for the volume of a sphere to find the volume.

Q: What is the volume of a sphere with a surface area of $2500 \pi \, \text{cm}^2$?

A: To find the volume of a sphere with a surface area of $2500 \pi \, \text{cm}^2$, you can first find the radius of the sphere using the formula for the surface area of a sphere:

$$4 \pi r^2 = 2500 \pi$$

Solving for $r$, you get:

$$r = 25$$

Then, you can use the formula for the volume of a sphere:

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

Substituting $r = 25$, you get:

$$V = \frac{4}{3} \pi (25)^3$$

Simplifying this expression, you get:

$$V = \frac{62500}{3} \pi$$

$$V = \frac{208331}{3} \pi \, \text{cm}^3$$

Conclusion

In this article, we answered some frequently asked questions related to the volume of a sphere with a given surface area. We provided formulas for the surface area and volume of a sphere, and we showed how to find the radius of a sphere with a given surface area. We also provided an example of how to calculate the volume of a sphere with a given surface area in terms of pi.