Geometry

Solids

Hard

Three small metallic cubes whose edges are in the ratio 3 : 4 : 5 are melted to form a big cube with no material lost. If the diagonal of the cube so formed is 18 cm, then the total surface area of the smallest cube (in $cm^2$ ) is

648

162

184

✅ Correct Option: 3

Since the edges are in the ratio $3:4:5$ , let the edges be $3k$ , $4k$ , and $5k$ .

Total volume of the three small cubes:

$= (3k)^3 + (4k)^3 + (5k)^3$

$= 27k^3 + 64k^3 + 125k^3$

$= 216k^3$

The big cube formed has volume $216k^3$ , so its side:

$= \sqrt[3]{216k^3}$

$= 6k$

The space diagonal of a cube with side $a$ is $a\sqrt{3}$ .

$6k\sqrt{3} = 18$

$k = \dfrac{18}{6\sqrt{3}}$

$k = \dfrac{3}{\sqrt{3}}$

$k = \sqrt{3}$

Smallest edge $= 3k = 3\sqrt{3}$ cm

Total surface area of the smallest cube:

$= 6 \times (3\sqrt{3})^2$

$= 6 \times 27$

$= 162 \text{ cm}^2$

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Three small metallic cubes whose edges are in the ratio 3 : 4 : 5 are melted to form a big cube with no material lost. If the diagonal of the cube so formed is 18 cm, then the total surface area of the smallest cube (in cm2cm^2cm2) is

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Three small metallic cubes whose edges are in the ratio 3 : 4 : 5 are melted to form a big cube with no material lost. If the diagonal of the cube so formed is 18 cm, then the total surface area of the smallest cube (in $cm^2$ ) is