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

Let the length $l$ ,width $w$ and height is $h$ of a box

Then total surface area $=A=2(lw+wh+lh)$

If the dimension is double the length is 2l ,width is 2w and height is 2h

Then new surface area of box=$2(2l\times 2w+2w\times 2h+2l\times 2h)=2(4lw+4wh+4lh)=8(lw+wh+lh)$

So the surface area when dimension doubled=$4\left[2\right(lw+lh+wh\left)\right]=4A=2^{2}A$

The change in surface area if dimension tripled =4 A =$2^{2}A$

Then if dimension is raised n times the surface area change =$n^{2}A$