Question

Each edge of a cube is increased by 50%. Find the percentage increase in the surface area of the cube.

Answer 1

Step I Given Terms

Percentage increase of edge of cube = 50%

Step II Finding the Edge of the Cube

Since, we know that each edge in a cube is same in length

Therefore, the edge of cube = x

Step III Finding the Initial Surface Area of Cube

Initial Surface area of the Cube = 6x²

Step III Finding the Percentage Increase in the Length of the Edge of Cube

According to the question, percentage increase in the edge of the cube = 50 %

Total percentage of the edge of the cube = 100%

Increase in the edge length of the cube = x + 50% of x

Upon simplification, we get

Increase in the edge length of the cube = $x+\frac{50}{100}x$

⇒ Increase in the edge length of the cube = $\frac{100x+50x}{100}$

= $$\begin{gathered} \frac{150x}{100} \\ =\frac{3x}{2} \end{gathered}$$

Step IV Finding the Surface area of Cube after Increase in Edge Length

Substituting the value of new edge length in Surface area of Cube, we get

Surface area of the Cube after increase in edge length = $6{\left(\frac{3x}{2}\right)}^2$

= $\frac{27}{2}{x}^2$

Step V Finding the Total Increase in Surface area of the Cube

Total increase in Surface area of the Cube = Initial Surface Area of the Cube - Surface Area of the Cube after 50% increase in edge length

On Substituting, the values of Surface areas in above equation, we get

Total increase in Surface area of the Cube = $\frac{27}{2}{x}^2-6x^2$

= $\frac{27x^2-12x^2}{2}$

= $\frac{15}{2}{x}^2$

Step VI Finding the Percentage increase in Surface Area of Cube

Percentage increase in Surface Area of Cube = $\frac{PercentageincreaseinSurfaceArea}{InitialSurfaceArea}\times 100$

On Substituting the Values, we get

Percentage Increase in Surface Area of Cube = $\frac{{\displaystyle \frac{15}{2}}{x}^2}{6x^2}\times 100$

= $\frac{15}{12}\times 100=125\%$

Therefore, Percentage Increase in Surface Area of Cube = 125%