Question

Each edge of a cube is increased by $\mathbf{50}\mathbf{\%}$. Find the percentage increase in the surface area of the cube.

Answer 1

Step 1: Find the new surface area of size increased cube

Let us consider the edge of the cube be “$a$”.

We know that

Total surface area of cube $=6a^2$

Given each edge of a cube is increased by $50\%$, then

New edge $=a+50\%$ of $a$

$=a+\frac{50}{100}a$

$=\frac{150}{100}a$

$\therefore$ New edge of a cube $=\frac{3}{2}a$

New surface area $=6{\left(\frac{3}{2}a\right)}^2$

$=6\times \frac{9}{4}{a}^2$

$\therefore$ New surface area $=\frac{27}{2}{a}^2$

Step 2: Find the increased value of surface area

Increased surface area $=$ new surface area $-$ surface area

$=\frac{27}{2}{a}^2-6a^2=\left(\frac{27-12}{2}\right)a^2$

$\therefore$ Increased surface area $=\frac{15}{2}{a}^2$

Step 3: Find the percentage of increased of surface area

Percentage of increase in surface area $=\left(\frac{Increasedsurfacearea}{orginalsurfacearea}\right)\times 100$

$=\left(\frac{{\displaystyle \frac{15}{2}}{a}^2}{6a^2}\right)\times 100$

$=\left(\frac{15a^2}{2}\times \frac{1}{6a^2}\right)\times 100$

$=\left(\frac{15}{12}\right)\times 100$

$\therefore$ Percentage of increase in surface area $=125\%$.

Hence, the percentage of increase in the surface area of the cube is $125\%$.