What Is The Volume Of A Sphere With A Surface Area Of $16 \pi \, \text{ft}^2$?A. $8 \pi \, \text{ft}^3$B. $10 \frac{2}{3} \pi \, \text{ft}^3$C. $21 \pi \, \text{ft}^3$D. $8 \pi^3 \, \text{ft}^3$

Introduction

In mathematics, the surface area and volume of a sphere are two fundamental properties that are often used to describe its size and shape. The surface area of a sphere is given by the formula $4 \pi r^2$, where $r$ is the radius of the sphere. On the other hand, the volume of a sphere is given by the formula $\frac{4}{3} \pi r^3$. In this article, we will explore the relationship between the surface area and volume of a sphere and use it 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 $4 \pi r^2$. This formula can be derived by considering the sphere as a collection of concentric circles, each with a radius equal to the radius of the sphere. The surface area of each circle is given by the formula $2 \pi r$, and since there are an infinite number of circles, the total surface area is given by the integral of $2 \pi r$ with respect to $r$. Evaluating this integral, we get $4 \pi r^2$.

Volume of a Sphere

The volume of a sphere is given by the formula $\frac{4}{3} \pi r^3$. This formula can be derived by considering the sphere as a collection of concentric spheres, each with a radius equal to the radius of the sphere. The volume of each sphere is given by the formula $\frac{4}{3} \pi r^3$, and since there are an infinite number of spheres, the total volume is given by the integral of $\frac{4}{3} \pi r^3$ with respect to $r$. Evaluating this integral, we get $\frac{4}{3} \pi r^3$.

Relationship Between Surface Area and Volume

The surface area and volume of a sphere are related by the following equation:

$$\text{Surface Area} = \frac{\text{Volume}}{r}$$

This equation can be derived by substituting the formulas for surface area and volume into the equation above. Simplifying the equation, we get:

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

Simplifying further, we get:

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

This equation shows that the surface area and volume of a sphere are proportional to each other.

Finding the Volume of a Sphere with a Given Surface Area

Now that we have established the relationship between the surface area and volume of a sphere, we can use it to find the volume of a sphere with a given surface area. Let's say we have a sphere with a surface area of $16 \pi \, \text{ft}^2$. We can use the formula for surface area to find the radius of the sphere:

$$4 \pi r^2 = 16 \pi$$

Simplifying the equation, we get:

$$r^2 = 4$$

Taking the square root of both sides, we get:

$$r = 2$$

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

$$\text{Volume} = \frac{4}{3} \pi r^3$$

Substituting the value of $r$ into the equation above, we get:

$$\text{Volume} = \frac{4}{3} \pi (2)^3$$

Simplifying the equation, we get:

$$\text{Volume} = \frac{4}{3} \pi (8)$$

Simplifying further, we get:

$$\text{Volume} = \frac{32}{3} \pi$$

Simplifying further, we get:

$$\text{Volume} = 10 \frac{2}{3} \pi \, \text{ft}^3$$

Therefore, the volume of a sphere with a surface area of $16 \pi \, \text{ft}^2$ is $10 \frac{2}{3} \pi \, \text{ft}^3$.

Conclusion

In this article, we have explored the relationship between the surface area and volume of a sphere. We have used this relationship to find the volume of a sphere with a given surface area. The formula for surface area is $4 \pi r^2$, and the formula for volume is $\frac{4}{3} \pi r^3$. The relationship between surface area and volume is given by the equation:

$$\text{Surface Area} = \frac{\text{Volume}}{r}$$

$$$$

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

A: The formula for the surface area of a sphere is $4 \pi r^2$, where $r$ 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 $\frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere.

Q: How is the surface area of a sphere related to its volume?

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

$$\text{Surface Area} = \frac{\text{Volume}}{r}$$

Q: How can 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 use the formula for surface area to find the radius of the sphere, and then use the formula for volume to find the volume of the sphere.

Q: What is the volume of a sphere with a surface area of $16 \pi \, \text{ft}^2$?

A: To find the volume of a sphere with a surface area of $16 \pi \, \text{ft}^2$, you can use the formula for surface area to find the radius of the sphere, and then use the formula for volume to find the volume of the sphere. The radius of the sphere is given by:

$$r^2 = \frac{\text{Surface Area}}{4 \pi}$$

Substituting the value of surface area, we get:

$$r^2 = \frac{16 \pi}{4 \pi}$$

Simplifying the equation, we get:

$$r^2 = 4$$

Taking the square root of both sides, we get:

$$r = 2$$

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

$$\text{Volume} = \frac{4}{3} \pi r^3$$

Substituting the value of $r$ into the equation above, we get:

$$\text{Volume} = \frac{4}{3} \pi (2)^3$$

Simplifying the equation, we get:

$$\text{Volume} = \frac{4}{3} \pi (8)$$

Simplifying further, we get:

$$\text{Volume} = \frac{32}{3} \pi$$

Simplifying further, we get:

$$\text{Volume} = 10 \frac{2}{3} \pi \, \text{ft}^3$$

Therefore, the volume of a sphere with a surface area of $16 \pi \, \text{ft}^2$ is $10 \frac{2}{3} \pi \, \text{ft}^3$.

Q: What is the volume of a sphere with a surface area of $32 \pi \, \text{ft}^2$?

A: To find the volume of a sphere with a surface area of $32 \pi \, \text{ft}^2$, you can use the formula for surface area to find the radius of the sphere, and then use the formula for volume to find the volume of the sphere. The radius of the sphere is given by:

$$r^2 = \frac{\text{Surface Area}}{4 \pi}$$

Substituting the value of surface area, we get:

$$r^2 = \frac{32 \pi}{4 \pi}$$

Simplifying the equation, we get:

$$r^2 = 8$$

Taking the square root of both sides, we get:

$$r = \sqrt{8}$$

Simplifying the equation, we get:

$$r = 2 \sqrt{2}$$

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

$$\text{Volume} = \frac{4}{3} \pi r^3$$

Substituting the value of $r$ into the equation above, we get:

$$\text{Volume} = \frac{4}{3} \pi (2 \sqrt{2})^3$$

Simplifying the equation, we get:

$$\text{Volume} = \frac{4}{3} \pi (8 \sqrt{2})$$

Simplifying further, we get:

$$\text{Volume} = \frac{32 \sqrt{2}}{3} \pi$$

Simplifying further, we get:

$$\text{Volume} = 21 \frac{2}{3} \pi \, \text{ft}^3$$

Therefore, the volume of a sphere with a surface area of $32 \pi \, \text{ft}^2$ is $21 \frac{2}{3} \pi \, \text{ft}^3$.

Q: What is the volume of a sphere with a surface area of $64 \pi \, \text{ft}^2$?

A: To find the volume of a sphere with a surface area of $64 \pi \, \text{ft}^2$, you can use the formula for surface area to find the radius of the sphere, and then use the formula for volume to find the volume of the sphere. The radius of the sphere is given by:

$$r^2 = \frac{\text{Surface Area}}{4 \pi}$$

Substituting the value of surface area, we get:

$$r^2 = \frac{64 \pi}{4 \pi}$$

Simplifying the equation, we get:

$$r^2 = 16$$

Taking the square root of both sides, we get:

$$r = 4$$

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

$$\text{Volume} = \frac{4}{3} \pi r^3$$

Substituting the value of $r$ into the equation above, we get:

$$\text{Volume} = \frac{4}{3} \pi (4)^3$$

Simplifying the equation, we get:

$$\text{Volume} = \frac{4}{3} \pi (64)$$

Simplifying further, we get:

$$\text{Volume} = \frac{256}{3} \pi$$

Simplifying further, we get:

$$\text{Volume} = 85 \frac{1}{3} \pi \, \text{ft}^3$$

Therefore, the volume of a sphere with a surface area of $64 \pi \, \text{ft}^2$ is $85 \frac{1}{3} \pi \, \text{ft}^3$.

Conclusion

In this article, we have answered some common questions about the volume of a sphere with a given surface area. We have used the formulas for surface area and volume to find the volume of spheres with surface areas of $16 \pi \, \text{ft}^2$, $32 \pi \, \text{ft}^2$, and $64 \pi \, \text{ft}^2$. We have found that the volume of a sphere with a surface area of $16 \pi \, \text{ft}^2$ is $10 \frac{2}{3} \pi \, \text{ft}^3$, the volume of a sphere with a surface area of $32 \pi \, \text{ft}^2$ is $21 \frac{2}{3} \pi \, \text{ft}^3$, and the volume of a sphere with a surface area of $64 \pi \, \text{ft}^2$ is $85 \frac{1}{3} \pi \, \text{ft}^3$.