A solid, metallic right circular cone of radius 7 cm and height 3 cm is melted into three cubes. If the sides of two cubes are 3 cm and 5 cm, then the side of the third cube will be how much?

$\sqrt[3]{2}$ cm

The correct answer is Option (2) → $\sqrt[3]{2}$ cm

$\text{Volume of cone} = \frac{1}{3} \pi r^2 h$

$= \frac{1}{3} \pi (7)^2 (3) = \frac{1}{3} \pi \cdot 49 \cdot 3 = 49\pi \ \text{cm}^3$

$\text{Volume of first cube (side = 3 cm)} = 3^3 = 27 \ \text{cm}^3$

$\text{Volume of second cube (side = 5 cm)} = 5^3 = 125 \ \text{cm}^3$

$\text{Let side of third cube be } x$

$\Rightarrow x^3 = 49\pi - 27 - 125 = 49\pi - 152$

Using $\pi \approx 3.14$:

$x^3 = 49 \cdot 3.14 - 152 = 153.86 - 152 = 1.86$

$\Rightarrow x = \sqrt[3]{1.86} \approx 1.22 \ \text{cm}$