Question

How does the total surface area of a box change if each dimension is doubled. Express in words. Can you find the area if each dimension is multiplied n times?

Answer 1

Hint: So, first of all, we all should know the basic formula of the cuboid where l is length, b is the breadth, \h’ is height and TSA is the total surface area of the cuboid as $TSA=2\left( lb+bh+hl \right)$. Then, the condition asked in the question is that we have to double the length, breadth, and height of the cuboid and then calculate the TSA again. Similarly, we can find the TSA for the condition if each side of the cuboid is multiplied n times. Then, by solving, we get the condition for the TSA of cuboid after doubling the sides and multiplying it n times.

Complete step-by-step solution:
In this question, we are supposed to find the total surface area of a box change if each dimension is doubled and also for the dimensions multiplied n times.
So, first of all, we all should know the basic formula of the cuboid where l is length, b is the breadth, ‘h’ is height and TSA is the total surface area of the cuboid shown below as:

So, now the TSA is given by the formula as:
$TSA=2\left( lb+bh+hl \right)$
Now, the condition asked in the question is that we have to double the length, breadth, and height of the cuboid and then calculate the TSA again.
So, let us assume the new TSA of the cuboid obtained by doubling the length, breadth and height is $TS{A}'$.
Now, by substituting all the values to get the TSA of the cuboid after doubling length, breadth and height is as:
$\begin{align}
  & TS{A}'=2\left[ \left( 2l\times 2b \right)+\left( 2b\times 2h \right)+\left( 2h\times 2l \right) \right] \\
 & \Rightarrow TS{A}'=4\times 2\left( lb+bh+hl \right) \\
 & \Rightarrow TS{A}'=4\times TSA \\
\end{align}$
So, we can conclude that if the length, breadth and height of the cuboid id doubled, then the TSA of the cuboid becomes 4 times of the original TSA.
Similarly, we can find the TSA for the condition if each side of the cuboid is multiplied n times.
Firstly, we will draw such a cuboid with dimension l, b and h multiplied n times as:

Now, the TSA for such a cuboid is named as $TS{A}''$ is given by:
$\begin{align}
  & TS{A}''=2\left[ \left( nl\times nb \right)+\left( nb\times nh \right)+\left( nh\times nl \right) \right] \\
 & \Rightarrow TS{A}''=2\times {{n}^{2}}\left( lb+bh+hl \right) \\
 & \Rightarrow TS{A}''={{n}^{2}}\times TSA \\
\end{align}$
Hence, if the dimensions of cuboid is multiplied n times, the total surface area become ${{n}^{2}}$times the original area.

Note: Now, in solving these types of problems, the only mistake we can occur is by confusing the curved surface area and total surface area of the cuboid which are different. As the curved surface area includes only the four faces of the cuboid excluding the bottom and upper face but total surface includes all faces. So, in this type of question always go with the total surface area. Then, we must know the formulas for the two as:
Curved surface area of cuboid is given by $2\left( l+b \right)h$.
Total surface of the cuboid id given by $2\left( lb+bh+hl \right)$.