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The length, breadth and height of a cuboid are in the ratio $5$:$4$:$2$. If the total surface area is $1216c{m^2}$, find the dimensions of the solid:
(A) $\left( {21 \times 11 \times 8} \right)c{m^3}$ (B) $\left( {20 \times 16 \times 8} \right)c{m^3}$ (C) $\left( {27 \times 17 \times 8} \right)c{m^3}$ (D) $\left( {25 \times 19 \times 8} \right)c{m^3}$

Last updated date: 13th Jul 2024
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Hint: Length, breadth and height are in the ratio $5$:$4$:$2$. Length will be $5x$, breadth will $4x$ and height will be $2x$. Use the formula of total surface area of cuboid to find the value of $x$.
Let the $l, b$ and $h$ be the length, breadth and height of the cuboid respectively. Then according to question:
 $ \Rightarrow $$l$:$b$:$h$$ = $ $5$:$4$:$2$.
Therefore let $l = 5x$, $b = 4x $and $h = 2x$.
Total surface area of the cuboid given in the question is $1216c{m^2}$. And we know that:
Total surface area of cuboid $ = 2\left( {lb + bh + lh} \right)$.
So, putting all values from above:
   \Rightarrow 2\left( {lb + bh + lh} \right) = 1216, \\
   \Rightarrow (5x)(4x) + (4x)(2x) + (5x)(2x) = 608, \\
   \Rightarrow 20{x^2} + 8{x^2} + 10{x^2} = 608, \\
   \Rightarrow 38{x^2} = 608, \\
   \Rightarrow {x^2} = 16, \\
   \Rightarrow x = 4. \\
Putting the value of $x$in $l,b$and$h$. We’ll get:
   \Rightarrow l = 5x = 20cm, \\
   \Rightarrow b = 4x = 16cm, \\
   \Rightarrow h = 2x = 8cm \\
Therefore, the dimensions of cuboid are $\left( {20 \times 16 \times 8} \right)c{m^3}$. (B) is the correct option.
Note: A cuboid consists of $6$ rectangular faces. Two of them have dimensions $(l \times b)c{m^2}$, another two have dimensions $(b \times h)c{m^2}$ and the rest two have dimensions\[(l \times h)c{m^2}\]. Therefore, the total surface area of the cuboid becomes $(2lb + 2bh + 2lh) = 2(lb + bh + lh)$ which we have used earlier. If all the dimensions $l,b$and$h$ are the same then it becomes a cube and in that case the total surface area is $6{l^2}$.