Question

The dimensions of a cuboid are in ratio \[4:3:2\]. If the total surface area of the cuboid is \[4212\] square metres. Find the volume of the cuboid.

Answer 1

Hint: Using the concept of ratio we know that if three numbers are in a ratio of \[a:b:c\] then they can be individually written as \[ak,{\text{ }}bk\] and \[ck\](where k is a constant of proportionality). Using these values in the formula of total surface area we find the value of the constant of proportionality and then find the volume.
* The total surface area of a Cuboid\[ = {\text{ }}2\left( {lb + bh + hl} \right)\]
* The volume of a cuboid$ = l \times b \times h$
where \[l,{\text{ }}b\] and \[h\] are length, breadth, and height respectively:

Complete step-by-step answer:
Express length, breadth and height in terms of k.
As it is given in the question that length, breadth and height are in the ratio of \[4:3:2\]
So, the length\[ = 4k\] … (1)
           Breadth\[ = 3k\] … (2)
              Height\[ = 2k\] ... (3)
Now we substitute the values in the formula for the total surface area and equate with the value given in the question to find the value of \[k\].
Since, total surface area of a Cuboid\[ = {\text{ }}2\left( {lb + bh + hl} \right)\]
Total Surface area given in the question\[ = {\text{ }}4212{m^2}\]
Equating both the values
\[2\left( {lb + bh + hl} \right) = 4212{m^2}\]
From (1), (2), and (3) we have
\[2\left( {4k \times 3k + 3k \times 2k + 2k \times 4k} \right) = 4212{m^2}\]
 $ \Rightarrow 2(12{k^2} + 6{k^2} + 8{k^2}) = 4212{m^2}$
$ \Rightarrow 52{k^2} = 4212{m^2}$
$ \Rightarrow {k^2} = 81$
Taking square root on both sides,
$
   \Rightarrow \sqrt {{k^2}} = \sqrt {81} \\
   \Rightarrow k = \sqrt {{9^2}} \\
   \Rightarrow k = \pm 9 \\
 $
Now since sides cannot be negative so k cannot have negative value
$ \Rightarrow k = 9$
Substitute the value of $k = 9$ in length, breadth, and height.
Length \[l = 4k = 4(9) = 36m\] … (4)
Breadth \[b = 3k = 3(9) = 27m\] ... (5)
Height \[h = 2k = 2(9) = 18m\] ... (6)
Now we find the volume by substituting the values in the formula.
We know volume $ = l \times b \times h$
                                 $ = 36 \times 27 \times 18$
                                 $ = 17496{m^3}$
The volume of the cuboid is $17496 {m^3}$

Note: Students are likely to make mistake while calculating the value of k as they might take the negative value also in perspective which is wrong because when substituted in length breadth and height k if negative will give negative values and all these are length measures which can never be negative.