The dimensions of a cuboid are in the ratio 5 : 3 : 1 and its total surface area is 414 m^{2}. Find the dimensions.

#### Solution

\[\text { It is given that the sides of the cuboid are in the ratio 5: 3: 1 } . \]

\[\text { Suppose that its sides are x multiple of each other, then we have: } \]

\[\text { Length = 5x m } \]

\[\text { Breadth = 3x m } \]

\[\text { Height = x m }\]

\[\text { Also, total surface area of the cuboid = 414 }m^2 \]

\[\text { Surface area of the cuboid = 2 }\times (\text { length } \times \text { breadth + breadth } \times \text { height + length }\times \text { height })\]

\[ \Rightarrow 414 = 2 \times (5x \times 3x + 3x \times 1x + 5x \times x)\]

\[ \Rightarrow 414 = 2 \times (15 x^2 + 3 x^2 + 5 x^2 ) \]

\[ \Rightarrow 414 = 2 \times (23 x^2 ) \]

\[ \Rightarrow 2 \times (23 \times x^2 ) = 414 \]

\[ \Rightarrow (23 \times x^2 ) = \frac{414}{2} = 207\]

\[ \Rightarrow x^2 =\frac{207}{23} = 9\]

\[ \Rightarrow x = \sqrt{9} = 3\]

\[\text { Therefore, we have the following: }\]

\[\text { Lenght of the cuboid = 5 } \times x = 5 \times 3 = 15 m \]

\[\text { Breadth of the cuboid = 3 } \times x = 3 \times 3 = 9 m \]

\[\text { Height of the cuboid = x = 1 } \times 3 = 3 m\]