The Volume Of A Cone Is $\frac{25}{3} \pi \, \text{cm}^3$. What Is The Volume Of A Sphere If Its Radius Is The Same As The Cone's And The Height Of The Cone Is Equal To The Sphere's Diameter?A. $25 \pi \, \text{cm}^3$ B.

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Introduction

In this problem, we are given the volume of a cone and asked to find the volume of a sphere under certain conditions. The volume of a cone is given by the formula $\frac{1}{3} \pi r^2 h$, where $r$ is the radius of the base and $h$ is the height of the cone. We are also given that the radius of the sphere is the same as the cone's radius and the height of the cone is equal to the sphere's diameter. We will use these conditions to find the volume of the sphere.

The Volume of a Cone

The volume of a cone is given by the formula $\frac{1}{3} \pi r^2 h$. In this problem, we are given that the volume of the cone is $\frac{25}{3} \pi \, \text{cm}^3$. We can set up an equation using this information:

$$\frac{1}{3} \pi r^2 h = \frac{25}{3} \pi$$

We can simplify this equation by canceling out the $\frac{1}{3} \pi$ term on both sides:

$$r^2 h = 25$$

The Relationship Between the Cone and the Sphere

We are given that the radius of the sphere is the same as the cone's radius and the height of the cone is equal to the sphere's diameter. This means that the diameter of the sphere is $h$, the height of the cone. We can use this information to find the radius of the sphere:

$$\text{diameter of sphere} = h$$

$$\text{radius of sphere} = \frac{h}{2}$$

The Volume of a Sphere

The volume of a sphere is given by the formula $\frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere. We can substitute the expression for the radius of the sphere in terms of $h$ into this formula:

$$\text{volume of sphere} = \frac{4}{3} \pi \left(\frac{h}{2}\right)^3$$

$$\text{volume of sphere} = \frac{4}{3} \pi \frac{h^3}{8}$$

$$\text{volume of sphere} = \frac{1}{6} \pi h^3$$

Finding the Volume of the Sphere

We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi h^3$$

$$\text{volume of sphere} = \frac{1}{6} \pi \left(\frac{25}{r^2}\right)^{\frac{3}{2}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{r^3}$$

We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{\left(\frac{25}{r^2}\right)^{\frac{3}{2}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{\frac{125}{r^3}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi r^3$$

Conclusion

We have found that the volume of the sphere is $\frac{1}{6} \pi r^3$. We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into this equation:

$$\text{volume of sphere} = \frac{1}{6} \pi r^3$$

$$\text{volume of sphere} = \frac{1}{6} \pi \left(\frac{25}{h}\right)^{\frac{3}{2}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{h^{\frac{3}{2}}}$$

We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{\left(\frac{25}{h}\right)^{\frac{3}{2}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{\frac{125}{h^{\frac{3}{2}}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi h^{\frac{3}{2}}$$

We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi h^{\frac{3}{2}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \left(\frac{25}{r^2}\right)^{\frac{3}{4}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{25}{r^{\frac{3}{2}}}$$

We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{25}{r^{\frac{3}{2}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{25}{\left(\frac{25}{r^2}\right)^{\frac{3}{4}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{25}{\frac{25}{r^{\frac{3}{2}}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi r^{\frac{3}{2}}$$

Final Answer

We have found that the volume of the sphere is $\frac{1}{6} \pi r^3$. We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into this equation:

$$\text{volume of sphere} = \frac{1}{6} \pi r^3$$

$$\text{volume of sphere} = \frac{1}{6} \pi \left(\frac{25}{h}\right)^{\frac{3}{2}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{h^{\frac{3}{2}}}$$

We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{\left(\frac{25}{h}\right)^{\frac{3}{2}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{\frac{125}{h^{\frac{3}{2}}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi h^{\frac{3}{2}}$$

We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi h^{\frac{3}{2}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \left(\frac{25}{r^2}\right)^{\frac{3}{4}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{25}{r^{\frac{3}{2}}}$$

We can substitute the expression for $r^2 h$ from the equation for the volume of the cone into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{25}{r^{\frac{3}{2}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{25}{\left(\frac{25}{r^2}\right)^{\frac{3}{4}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{25}{\frac{25}{r^{\frac{3}{2}}}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi r^{\frac{3}{2}}$$

The final answer is: $\boxed{25 \pi}$

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Q: What is the relationship between the cone and the sphere?

A: The radius of the sphere is the same as the cone's radius, and the height of the cone is equal to the sphere's diameter.

Q: How do we find the volume of the sphere?

A: We can use the formula for the volume of a sphere, which is $\frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere. We can substitute the expression for the radius of the sphere in terms of $h$ into this formula.

Q: What is the expression for the radius of the sphere in terms of $h$?

A: The radius of the sphere is $\frac{h}{2}$.

Q: How do we find the volume of the sphere in terms of $h$?

A: We can substitute the expression for the radius of the sphere in terms of $h$ into the formula for the volume of a sphere:

$$\text{volume of sphere} = \frac{4}{3} \pi \left(\frac{h}{2}\right)^3$$

$$\text{volume of sphere} = \frac{4}{3} \pi \frac{h^3}{8}$$

$$\text{volume of sphere} = \frac{1}{6} \pi h^3$$

Q: How do we find the value of $h$?

A: We can use the equation for the volume of the cone, which is $\frac{1}{3} \pi r^2 h = \frac{25}{3} \pi$. We can simplify this equation by canceling out the $\frac{1}{3} \pi$ term on both sides:

$$r^2 h = 25$$

Q: How do we find the value of $r$?

A: We can use the equation for the volume of the cone, which is $\frac{1}{3} \pi r^2 h = \frac{25}{3} \pi$. We can simplify this equation by canceling out the $\frac{1}{3} \pi$ term on both sides:

$$r^2 h = 25$$

Q: How do we find the value of the volume of the sphere?

A: We can substitute the expression for $h$ in terms of $r$ into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi h^3$$

$$\text{volume of sphere} = \frac{1}{6} \pi \left(\frac{25}{r^2}\right)^{\frac{3}{2}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{r^3}$$

Q: What is the final answer?

A: The final answer is $\boxed{25 \pi}$.

Q: What is the relationship between the cone and the sphere?

A: The radius of the sphere is the same as the cone's radius, and the height of the cone is equal to the sphere's diameter.

Q: How do we find the volume of the sphere?

A: We can use the formula for the volume of a sphere, which is $\frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere. We can substitute the expression for the radius of the sphere in terms of $h$ into this formula.

Q: What is the expression for the radius of the sphere in terms of $h$?

A: The radius of the sphere is $\frac{h}{2}$.

Q: How do we find the volume of the sphere in terms of $h$?

A: We can substitute the expression for the radius of the sphere in terms of $h$ into the formula for the volume of a sphere:

$$\text{volume of sphere} = \frac{4}{3} \pi \left(\frac{h}{2}\right)^3$$

$$\text{volume of sphere} = \frac{4}{3} \pi \frac{h^3}{8}$$

$$\text{volume of sphere} = \frac{1}{6} \pi h^3$$

Q: How do we find the value of $h$?

A: We can use the equation for the volume of the cone, which is $\frac{1}{3} \pi r^2 h = \frac{25}{3} \pi$. We can simplify this equation by canceling out the $\frac{1}{3} \pi$ term on both sides:

$$r^2 h = 25$$

Q: How do we find the value of $r$?

A: We can use the equation for the volume of the cone, which is $\frac{1}{3} \pi r^2 h = \frac{25}{3} \pi$. We can simplify this equation by canceling out the $\frac{1}{3} \pi$ term on both sides:

$$r^2 h = 25$$

Q: How do we find the value of the volume of the sphere?

A: We can substitute the expression for $h$ in terms of $r$ into the equation for the volume of the sphere:

$$\text{volume of sphere} = \frac{1}{6} \pi h^3$$

$$\text{volume of sphere} = \frac{1}{6} \pi \left(\frac{25}{r^2}\right)^{\frac{3}{2}}$$

$$\text{volume of sphere} = \frac{1}{6} \pi \frac{125}{r^3}$$

Q: What is the final answer?

A: The final answer is $\boxed{25 \pi}$.