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?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\sqrt[3]{2}\) cm
When a solid is melted and recast, its volume does not change, so the volume of the cone equals the total volume of the three cubes.
The volumes (the cone equals the three cubes together):
Volume of the cone \(\displaystyle =\frac{1}{3}\pi r^2h=\frac{1}{3}\times\frac{22}{7}\times7\times7\times3=154\,\mathrm{cm^3}\)
First known cube: \(3^3=27\,\mathrm{cm^3}\)
Second known cube: \(5^3=125\,\mathrm{cm^3}\)
The two known cubes together: \(152\,\mathrm{cm^3}\)
The third cube (what is left of the cone):
Volume of the third cube \(=154-152=2\,\mathrm{cm^3}\)
Its side is the cube root of its volume: \(\sqrt[3]{2}\) cm