Ratio Between the Height of a Cone and the Diameter of a Sphere with Equal Radii and Volumes

In this article, we will explore the geometric relationship between a cone and a sphere when they have equal radii and volumes. Specifically, we will derive the ratio between the height of the cone and the diameter of the sphere. This exploration is crucial for understanding the interplay of dimensions in three-dimensional shapes and can be a valuable resource for students of geometry and mathematics.

Introduction to Volumes and Geometric Shapes

The problem at hand involves two fundamental geometric shapes: a cone and a sphere. Each shape has a distinct relationship between its dimensions and volume.

Volume of a Cone

The volume of a cone is given by the formula:

$V_{text{cone}} frac{1}{3} pi r^2 h$

where $r$ is the radius of the base of the cone and $h$ is the height of the cone.

Volume of a Sphere

The volume of a sphere is given by the formula:

$V_{text{sphere}} frac{4}{3} pi r^3$

where $r$ is the radius of the sphere.

Setting Equal Volumes

Given that the cone and the sphere have equal volumes and radii, we set their volume equations equal to each other:

$frac{1}{3} pi r^2 h frac{4}{3} pi r^3$

To simplify this equation, we can cancel out the common terms on both sides:

$r^2 h 4r^3$

Solving for Height of the Cone

Dividing both sides by $r^2$ (assuming $r eq 0$), we find the height of the cone:

$h 4r$

Calculating the Diameter of the Sphere

The diameter of a sphere is given by:

$d 2r$

Deriving the Ratio

Now, to find the ratio between the height of the cone and the diameter of the sphere, we divide the height of the cone by the diameter of the sphere:

$frac{h}{d} frac{4r}{2r} 2$

Thus, the ratio between the height of the cone and the diameter of the sphere is:

$boxed{2}$

Conclusion

This exploration has demonstrated the relationship between the dimensions of a cone and a sphere when their volumes and radii are equal. The ratio of 2 highlights the proportional relationship between the height of the cone and the diameter of the sphere in such scenarios.