Question

The surface area of a cuboid is $1372c{m^2}$. If its dimensions are in the ratio of $4:2:1$. Find the length.

Answer 1

Hint:
We will suppose that the dimensions of the cuboid will be 4k, 2k and 1k since they are in the ratio $4:2:1$. The area of the cuboid is given by $2\left( {lb + bh + hl} \right)$ where, l is the length of the cuboid, b is the breadth of the cuboid and h is the height of the cuboid. We will put the values and equate it to the given surface area of the cuboid in order to determine the value of k and hence the value of length l.

Complete step by step solution:
We are given that the surface area of a cuboid is $1372c{m^2}$.
The dimensions of the cuboid are given in a ratio as: $4:2:1$
Let k be any constant natural number such that the dimensions of the cuboid becomes: length = 4k, breadth = 2k and height = 1k = k.
Figure:

We know that the surface area of the cuboid is given by: $2\left( {lb + bh + hl} \right)$ where, l is the length of the cuboid, b is the breadth of the cuboid and h is the height of the cuboid.
Substituting the values of l, b and h, we get
$ \Rightarrow $ Surface area of the cuboid = $2\left( {lb + bh + hl} \right)$
$ \Rightarrow $ Surface area of the cuboid =$2\left[ {\left( {4k} \right)\left( {2k} \right) + \left( {2k} \right)\left( {1k} \right) + \left( {1k} \right)\left( {4k} \right)} \right]$
$ \Rightarrow $ Surface area of the cuboid = $2\left[ {8{k^2} + 2{k^2} + 4{k^2}} \right] = 2\left[ {14{k^2}} \right] = 28{k^2}$
We are given in the question that the surface area of the cuboid is$1372c{m^2}$
$ \Rightarrow $ Surface area of the cuboid = $1372c{m^2} = 28{k^2}$
$ \Rightarrow {k^2} = \dfrac{{1372}}{{28}} = 49$
$ \Rightarrow k = \sqrt {49} = 7$
Now, we know the value of k, so we can easily calculate the value of l, b and h by putting the value of k.

Therefore, the length of the cuboid is $4k = 4\left( 7 \right) = 28cm$

Note:
In this question, you need not to calculate the values of height and breadth by using the value of k as the question was only about the value of the length of the cuboid. You may go wrong while calculating the value of k using the formula of the surface area of the cuboid.