Question

A solid cube is cut into 27 identical cubes . What is the percentage increase in the total surface area

Answer 1

So lets say the side length of the original cube is 'x'.

Now the volume of the original cube is equal to $x^3$

and the surface area of the original cube $=6x^2$

If the cube is cut into 27 identical cubes then the volume of each cube is:

$x_n^3=\frac{x^3}{27}$

where, $x_n$ is the side length of each smaller cube. So, the surface area of each tiny cube is:

$6x_n^2=6\times (\frac{x}{3} )^2=6\times\frac{x^2}{9} =\frac{2x^2}{3}$

Now, there are 27 identical cubes so the total surface area is:

$27\times \frac{2x^2}{3} =18x^2$

Now, the surface area of the original cube is $6x^2$

and the total surface area of 27 identical cubes is $18x^2$

So, the percentage increase in the total surface area is:

Percentage increase $=\frac{N-O}{O} \times 100\%$, where N stands for new value of the surface area, O stands for the old value of the surface area.

$\frac{18x^2-6x^2}{6x^2}\times 100 \% =\frac{12x^2}{6x^2}\times 100\% =2\times 100 \%=200\%$

So the percentage increase in the total surface area is 200%.