Question

The rectangle in the figure has sides 2a and 2b ; the curve is an ellipse. If the figure is rotated about the dotted line it generates three solids of revolution: a cone, an ellipsoid, and a cylinder. Show that the volumes are in the ratio 1:2:3 . (See L.H. Lange, American MathematicalMonthly vol. 88 (1981), p. 339.)

Short answer

The volumes are in the ratio 1:2:3 .

Step-by-step solution

The volumes are in the ratio .

The sides of the rectangle are 2a,2b.

Step 2: Concept of the volumes

The volumes of the cone and cylinder are straightforward, and the volume of the ellipsoid can be found by transforming into a system where it turns into a sphere.

Find the volume of the cylinder

The volume of the cylinder is as follows:

$$\begin{gathered} {\mathrm{V}}_1=\left({\mathrm{\tau \tau a}}^2\right)\left(2\mathrm{b}\right) \\ =2{\mathrm{\tau \tau a}}^2\mathrm{b} \end{gathered}$$

Find the volume of the ellipsoid and transform it as follows:

$$\begin{gathered} x=x\text{'}a \\ y=y\text{'}b \\ z=z\text{'}b \end{gathered}$$

$$\begin{gathered} {\mathrm{V}}_2={\int }_{\mathrm{x}}{\int }_{\mathrm{y}}{\int }_{\mathrm{z}}\mathrm{dx}\text{'}\mathrm{dy}\text{'}\mathrm{dz}\text{'}{\mathrm{a}}^2\mathrm{b} \\ ={\mathrm{a}}^2\mathrm{b}\frac{4\mathrm{\tau \tau }}{3} \\ =\frac{4\mathrm{\tau \tau }}{3}{\mathrm{a}}^2\mathrm{b} \end{gathered}$$

Where in the transformed equation ellipsoid turns into a sphere of radius 1.

Find the volume of the cone

The volume of the cone can also be found by transforming:

$$\begin{gathered} {\mathrm{V}}_3={\mathrm{a}}^2\mathrm{b}{\int }_{\mathrm{o}}^{2\mathrm{\tau \tau }}\mathrm{d\theta }{\int }_{-1}^1\mathrm{dz}\text{'}{\int }_0^1\mathrm{rdr} \\ =2{\mathrm{\tau \tau a}}^2\mathrm{b}{\int }_{-1}^1\mathrm{dz}\frac{1}{2}{\mathrm{z}}^2 \\ =\frac{2\mathrm{\tau \tau }}{3}{\mathrm{a}}^2\mathrm{b} \end{gathered}$$

Origin at the center of the ellipsoid.

Thus, proving the theorem results in the following,

$$\begin{gathered} \frac{V_2}{V_3}=2 \\ \frac{V_1}{V_3}=3 \end{gathered}$$

Hence, the volumes are in the ratio 1:2:3 .