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

Answer

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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.

Complete step-by-step answer:

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.

Complete step-by-step answer:

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.

Last updated date: 31st May 2023

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