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The surface of cuboid is $4212\text{ }{{\text{m}}^{\text{2}}}$ and if its dimension are in the ratio $4:3:2$ then find the volume.

Last updated date: 20th Jun 2024
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Hint: First we will define what is meant by cuboid then we will assume a variable for the length breadth and the height of the cuboid and then we will apply the formula for the total surface area of a cuboid that is $2\left( lb+bh+hl \right)$ then we will get the dimensions of the cuboid. Then for finding the volume of the cuboid we will apply the formula: $\left( length\times breadth\times height \right)$and get the answer.

Complete step by step answer:
Let’s first see what is meant by a cuboid. So, A cuboid is a three dimensional shape having six faces, eight vertices and twelve edges. The faces of the cuboid are parallel. But not all the faces of a cuboid are equal in dimensions.
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Now let the length, breadth and height of the cuboid be $4x,3x\text{ and }2x$.
Given that the surface area of the cuboid is $4212\text{ }{{\text{m}}^{\text{2}}}$.
We know that the surface area of a cuboid of length $l$ , breadth $b$ and height $h$ is $2\left( lb+bh+hl \right)$
Now, we will put the values given in the question to find out the surface area of cuboid:
  & \Rightarrow 4212=2\left( \left( 4x.3x \right)+\left( 3x.2x \right)+\left( 2x.4x \right) \right) \\
 & \Rightarrow 2106=\left( 12{{x}^{2}}+6{{x}^{2}}+8{{x}^{2}} \right) \\
 & \Rightarrow 2106=26{{x}^{2}} \\
 & \Rightarrow {{x}^{2}}=81 \\
Taking square roots on both the sides, we will get: $\Rightarrow x=\pm 9$ , since dimension cannot be negative therefore: $x=9$
Now, the length of the cuboid will be $4x=4\times 9=36\text{ m}$ , breadth will be $3x=3\times 9=27\text{ m}$ and height $2x=2\times 9=18\text{ m}$
Now, we know that the volume of the cuboid is $\left( length\times breadth\times height \right)$
Therefore, the volume of the given cuboid will be: $\left( 36\times 27\times 18 \right)=17496\text{ }{{\text{m}}^{3}}$

Hence, the answer is $17496\text{ }{{\text{m}}^{3}}$ .

Note: Common mistake can be made while applying the total surface area formula and instead of that one can use the lateral surface area formula that is $2h\left( l+b \right)$, this will lead to a totally different answer. Also, units must be mentioned at every step.