Question

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

Answer 1

Hint: In this problem, first we need to find the surface area of the cube with increased in edges of the cube. Next, find the percentage increase in surface area.

Complete step-by-step solution:
The surface \[S\] are of a cube having edge length \[a\] is as follows.

\[S = 6{a^2}\]
Since, the edge length of the cube is increased by 50%, therefore
${a_1} = \left( {a + \dfrac{{50}}{{100}} \times a} \right)$
 $ \Rightarrow {a_1} = \left( {a + 0.5a} \right) $
 $\Rightarrow {a_1} = 1.5a $
Substitute, \[1.5a\] for \[a\] in the above surface area formula to obtain the new surface area \[{S_1}\] of the cube.
  ${S_1} = 6{\left( {1.5a} \right)^2} $
 $\Rightarrow {S_1} = 6\left( {2.25{a^2}} \right) $
The percentage increase in the surface area of the cube is calculated as shown below.
The percentage increase in the surface area = $\dfrac{{{S_1} - S}}{S} \times 100$
Substitute, \[6{a^2}\] for\[S\], \[6\left( {2.25{a^2}} \right)\] for \[{S_1}\] in above formula.
The percentage increase in the surface area = $\dfrac{{6\left( {2.25{a^2}} \right) - 6{a^2}}}{{6{a^2}}} \times 100 $
The percentage increase in the surface area = $\dfrac{{6{a^2}\left( {2.25 - 1} \right)}}{{6{a^2}}} \times 100 $
The percentage increase in the surface area = $1.25 \times 100$
The percentage increase in the surface area =$ 125 $
Thus, the percentage increase in surface area of the cube is 125, hence, option (B) is the correct answer.

Note: Here the mean of percentage changes in surface area is changing in the area of a cube on increasing the side. In the percentage increase formula, we divide by the initial surface area instead of new surface area as the changes are always to be found in respect of initial value. There are six identical faces of a cube having an area equal to the side’s square.