Question

The volume of a cuboid is $24\sqrt{42}{\mathrm{cm}}^3$. Its length is $5\sqrt{2}$ cm, breadth and height are in the ratio $\sqrt{3}:\sqrt{7}$. The height of the cuboid is ________.

Answer 1

Let the breadth and height of the cuboid be $\sqrt{3}x$ and $\sqrt{7}x$, respectively.

Volume of the cuboid = Length × Breadth × Height

$$\begin{gathered} \therefore 5\sqrt{2}\times \sqrt{3}x\times \sqrt{7}x=24\sqrt{42}{\mathrm{cm}}^3 \\ \Rightarrow 5\sqrt{42}{x}^2=24\sqrt{42} \\ \Rightarrow x^2=\frac{24}{5} \\ \Rightarrow x=\sqrt{\frac{24}{5}}\mathrm{cm} \end{gathered}$$

∴ Height of the cuboid = $\sqrt{7}x=\sqrt{7}\times \sqrt{\frac{24}{5}}=\frac{2}{5}\sqrt{210}\mathrm{cm}$

The volume of a cuboid is $24\sqrt{42}{\mathrm{cm}}^3$. Its length is $5\sqrt{2}$ cm, breadth and height are in the ratio $\sqrt{3}:\sqrt{7}$. The height of the cuboid is $\underline{\frac{2}{5}\sqrt{210}\mathrm{cm}}$.