Question

# The dimensions of a cuboid are in the ratio 1:2:3 and its total surface area is 88${m^2}$. Find the dimensions.

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Hint: The relation between the dimensions of the cuboid is given in the question along with its total surface area. So let any one side of the cuboid be a variable and use the provided relations to calculate this variable, and thus other sides can be obtained too.

It is given that the total surface area of the cuboid is $88{\text{ }}{{\text{m}}^2}$.
And the dimensions of the cuboid is in the ratio 1:2:3
Let one side of the cuboid be x m.
Then according to their dimensions ratios other sides are 2x and 3x m respectively.
Let the length (l) of the cuboid = x meter.
The breadth (b) of the cuboid = 2x meter.
And the height (h) of the cuboid = 3x meter.
Now as we know cuboid has six faces so, the total surface area (S.A) of the cuboid is
$S.A = 2\left( {lb + bh + hl} \right)$
Now, substitute the values in above equation we have,
$\Rightarrow S.A = 2\left[ {\left( {x \times 2x} \right) + \left( {2x \times 3x} \right) + \left( {3x \times x} \right)} \right]$ Sq. meter
Now simplify it we have,
$\Rightarrow S.A = 2\left[ {2{x^2} + 6{x^2} + 3{x^2}} \right] = 22{x^2}$ Sq. meter
But it is given that the total surface area is$88{\text{ }}{{\text{m}}^2}$.
So, equate them we have,
$\Rightarrow S.A = 22{x^2} = 88$
$\Rightarrow {x^2} = \dfrac{{88}}{{22}} = 4$
$\Rightarrow x = \sqrt 4 = 2$ m
Therefore the length (l) of the cuboid = 2 meter.
The breadth (b) of the cuboid = 4 meter.
And the height (h) of the cuboid = 6 meter.
So, the dimensions of the cuboid are 2 m, 4 m and 6m.
So, this is the required answer.

Note: Whenever we face such types of problems the key point is to have a good grasp over some of the basic formulas like Total surface area of a cuboid or other conic sections. The understanding of these formulas always help in simplifying the problem and thus getting you on the right track to reach the answer.