Question

a bigger cube is formed from the material obtained by melting tree smaller cubes of 3cm, 4cm, 5cm sides. whatis the ratio of the total surface area of the smaller cubes and the bigger cubes

Answer 1

Sides of the three smaller cubes are - 3cm, 4cm and 5cm.

If we melt the cubes and make a big one, the volume of the big cube will be equal to sum of volumes of the three smaller cubes.

Total volume of three cubes -

${3}^{3} + {4}^{3} + {5}^{3} \\ = 27 + 64 + 125 \\ = 216 {cm}^{3}$

Side of the the big cube -
$\sqrt[3]{volume} \\ = \sqrt[3]{216} \\ = 6 cm$
Now the total TSA of the three smaller cubes -
$6 \times {3}^{2} + 6 \times {4}^{2} + 6 \times {5}^{2} \\ = 6( {3}^{2} + {4}^{2} + {5}^{2} ) \\ = 6(9 + 16 + 25) \\ = 6(50) \\ = 300{cm}^{2}$
TSA of the bigger cube -
$6 \times {6}^{2} \\ = 6 \times 36 \\ = 216 {cm}^{2}$
Ratio of the total TSA of smaller cubes to TSA of bigger cube -
$\frac{300}{216 } \\ = \frac{25}{18}$
Answer - The ratio of the TSA of smaller cubes to that of bigger cube is 25:18.