The Ratio of Volumes of a Cone, Cylinder, and Hemisphere with Equal Heights

In this article, we will explore the problem of determining the ratio of the volumes of a cone, cylinder, and hemisphere when their heights and radii are given in a specific ratio. Specifically, we will consider a scenario where the heights of a cone, cylinder, and hemisphere are equal, and the radii of these three shapes are in the ratio 2:3:1. We will derive the volumes and then find the ratio of these volumes.

Formulas for Volumes

The volume of a cone is given by:

Vco (frac{1}{3} pi r^2 h)

The volume of a cylinder is given by:

Vcy (pi r^2 h)

The volume of a hemisphere is given by:

Vhm (frac{2}{3} pi r^3 cdot frac{1}{2} frac{2}{3} pi r^3)

Given Information

Let the following be the radius and height of the relevant shapes:

Radius of the cone, (r_1 2r) Radius of the cylinder, (r_2 3r) Radius of the hemisphere, (r_3 r) Heights of the shapes, (h_1 h_2 h_3 h)

Calculation of Volumes and Ratios

Since the heights of the cone, cylinder, and hemisphere are equal, we can substitute (h) for the height in the volume formulas. Let's proceed to calculate the volumes step-by-step.

1. Vco (frac{1}{3} pi (2r)^2 h frac{1}{3} pi cdot 4r^2 h frac{4}{3} pi r^2 h)

2. Vcy (pi (3r)^2 h pi cdot 9r^2 h 9 pi r^2 h)

3. Vhm (frac{2}{3} pi r^3)

Since the radius (r_3 r h), we substitute (r^3) for the hemisphere's radius, resulting in:

Vhm (frac{2}{3} pi r^3)

Now, we can express the volumes as:

Vco : Vcy : Vhm (frac{4}{3} pi r^2 h : 9 pi r^2 h : frac{2}{3} pi r^3)

To simplify, we can divide each term by (pi r^2 h), which gives:

4: 27: (frac{2r}{3h})

However, since (r h) in the case of the hemisphere, the expression simplifies further to:

4: 27: (frac{2}{3})

Final Ratio

Thus, the ratio of the volumes of the cone, cylinder, and hemisphere is 4:27:2.

Conclusion

We have derived the ratio of the volumes of a cone, cylinder, and hemisphere when their radii are in the ratio 2:3:1 and their heights are equal. The final ratio of the volumes is 4:27:2. This detailed step-by-step calculation helps in understanding the geometric relationships and the importance of consistent units and ratios in solving such problems in geometry and calculus.