Question

The dimensions of a cuboid are in the ratio 5 : 3 : 1 and its total surface area is 414 m². Find the dimensions.

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{the}\mathrm{sides}\mathrm{of}\mathrm{the}\mathrm{cuboid}\mathrm{are}\mathrm{in}\mathrm{the}\mathrm{ratio}5:3:1. \\ \mathrm{Suppose}\mathrm{that}\mathrm{its}\mathrm{sides}\mathrm{are}x\mathrm{multiple}\mathrm{of}\mathrm{each}\mathrm{other},\mathrm{then}\mathrm{we}\mathrm{have}: \\ \mathrm{Length}=5x\mathrm{m} \\ \mathrm{Breadth}=3x\mathrm{m} \\ \mathrm{Height}=x\mathrm{m} \\ \mathrm{Also},\mathrm{total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cuboid}=414{\mathrm{m}}^2 \\ \mathrm{Surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cuboid}=2\times (\mathrm{length}\times \mathrm{breadth}+\mathrm{breadth}\times \mathrm{height}+\mathrm{length}\times \mathrm{height}) \\ \Rightarrow 414=2\times (5x\times 3x+3x\times 1x+5x\times x) \\ \Rightarrow 414=2\times (15x^2+3x^2+5x^2) \\ \Rightarrow 414=2\times \left(23x^2\right) \\ \Rightarrow 2\times (23\times x^2)=414 \\ \Rightarrow (23\times x^2)=\frac{414}{2}=207 \\ \Rightarrow x^2\mathrm{=}\frac{207}{23}=9 \\ \Rightarrow x=\sqrt{9}=3 \\ \mathrm{Therefore},\mathrm{we}\mathrm{have}\mathrm{the}\mathrm{following}: \\ \mathrm{Lenght}\mathrm{of}\mathrm{the}\mathrm{cuboid}=5\times \mathrm{x}=5\times 3=15\mathrm{m} \\ \mathrm{Breadth}\mathrm{of}\mathrm{the}\mathrm{cuboid}=3\times \mathrm{x}=3\times 3=9\mathrm{m} \\ \mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{cuboid}=\mathrm{x}=1\times 3=3\mathrm{m} \end{gathered}$$