Question

If a solid cube of total surface area 'S is cut into 64 identical cubes, by what value
would the total surface area increase?​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

Let A be the side of the big solid cube

Also let a be the side of the each small cube

Since the solid cube is cut into 64 identical cubes

Then Volume of the big solid cube = 64 × Volume of each small cube

So

$\sf{ {A}^{3} = 64 {a}^{3} }$

$\implies \: \sf{ {A}^{3} = {(4a)}^{3} }$

$\implies \: \sf{ {A} = 4a }$

Now the total surface area of the big solid cube

$= \sf{ 6 {A}^{2} \: }$

$= \sf{ \:6 \times {(4a)}^{2} \: }$

$= \sf{ \:96 {a}^{2} \: }$

Now the total surface area of each small cube

$= \sf{ 6 {a}^{2} \: }$

So the total surface area of 64 small cubes

$= \sf{ 64 \times 6 {a}^{2} \: }$

$= \sf{ 384 {a}^{2} \: }$

So the increase in total surface area

$= \sf{ 384 {a}^{2} - 96 {a}^{2} \: }$

$= \sf{288{a}^{2} }$

So the required rate of increase

$= \displaystyle \sf{ \: \frac{288 {a}^{2} }{96 {a}^{2} } \times 100 } \%$

$= \sf{ \: 300 \: } \%$

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