Question

How does the total surface area of a box change if

Each dimension is tripled?
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 3l ,width is 3w and height is 3h

Then new surface area of box=$2(3l\times 3w+3w\times 3h+3l\times 3h)=2(4lw+4wh+4lh)=18(lw+wh+lh)$

So the surface area when dimension tripled=$9\left[2\right(lw+lh+wh\left)\right]=9A=3^{2}A$

The change in surface area if dimension tripled =9 A =$3^{2}A$

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