Question

If each edge of a cube is increased by 50%, then the percentage increase in the cube surface area is:
$
  {\text{A}}{\text{. 50}} \\
  {\text{B}}{\text{. 125}} \\
  {\text{C}}{\text{. 150}} \\
  {\text{D}}{\text{. 300}} \\
$

Answer 1

Hint – To find the percentage increase, we find the new length of the edge of the cube. Then we find the new surface area and use the percentage formula.

Complete Step-by-Step solution:
Let the length of each edge of the cube = a cm

We know Surface area of a cube of a side length a is given by S = $6{{\text{a}}^2}$

Given Data: each edge of a cube is increased by 50%.

⟹new edge length of the cube = a + a × $\dfrac{{50}}{{100}}$
                                                         = a + $\dfrac{{\text{a}}}{2}$
                                                         = $\dfrac{{{\text{3a}}}}{2}$

New surface area of the cube = 6 x ${\left( {\dfrac{{{\text{3a}}}}{2}} \right)^2}$

We know percent increase is given by,
Percentage increase = $\dfrac{{{\text{new quantity - old quantity}}}}{{{\text{old quantity}}}} \times 100$

Hence, Percentage increase in the cube surface area is given by,
$ \Rightarrow \dfrac{{{\text{new surface area - old surface area}}}}{{{\text{old surface area}}}} \times 100$
$
   \Rightarrow \dfrac{{\left( {6 \times {{\left( {\dfrac{{3{\text{a}}}}{2}} \right)}^2}} \right) - 6{{\text{a}}^2}}}{{{\text{6}}{{\text{a}}^2}}} \times 100 \\
   \Rightarrow \dfrac{{\left( {6 \times \left( {\dfrac{{{\text{9}}{{\text{a}}^2}}}{4}} \right)} \right) - 6{{\text{a}}^2}}}{{{\text{6}}{{\text{a}}^2}}} \times 100 \\
   \Rightarrow \left( {\dfrac{9}{4} - 1} \right) \times 100 \\
$
$
   \Rightarrow \dfrac{5}{4} \times 100 \\
   \Rightarrow 125\% \\
$

Therefore, the percentage increase in the cube surface area is = 125%

Note: The key in solving such types of problems is to know the formula of surface area of a cube. Finding the percentage increase in the cube surface area is crucial.
Percentage is a number or a ratio that represents a fraction of 100.
Percentage increase is the division of difference of percentages and original percentage multiplied by 100.