Question

The ratio of the volumes of two cubes is 729 : 1331. The ratio of their total surface areas is __________.

Answer 1

Let the edges of the two cubes be x units and y units.

Volume of cube 1 : Volume of cube 2 = 729 : 1331 (Given)

$\therefore \frac{x^3}{y^3}=\frac{729}{1331}$ [Volume of the cube = (Edge)³]

$\Rightarrow {\left(\frac{x}{y}\right)}^3={\left(\frac{9}{11}\right)}^3$

$\Rightarrow \frac{x}{y}=\frac{9}{11}.....\left(1\right)$

$\therefore \frac{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{cube}1}{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{cube}2}=\frac{6x^2}{6y^2}$

$\Rightarrow \frac{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{cube}1}{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{cube}2}={\left(\frac{x}{y}\right)}^2={\left(\frac{9}{11}\right)}^2=\frac{81}{121}$ [Using (1)]

⇒ Total surface area of cube 1 : Total surface area of cube 2 = 81 : 121

Thus, the ratio of their total surface areas is 81 : 121.

The ratio of the volumes of two cubes is 729 : 1331. The ratio of their total surface areas is ___81 : 121___.