To solve this problem, we need to follow several steps. Let's break it down:
1. Calculate the volume of the metal cuboid.
- The dimensions of the cuboid are given as 12 cm in length, 16 cm in width, and 22 cm in height.
- The volume [tex]\( V \)[/tex] of a cuboid is calculated using the formula:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} \][/tex]
- Plugging in the given values:
[tex]\[ V = 12 \times 16 \times 22 = 4224 \, \text{cm}^3 \][/tex]
2. Convert this volume into a cube.
- When the cuboid is melted and formed into a cube, the volume of the cube will be the same as the volume of the cuboid, which is [tex]\( 4224 \, \text{cm}^3 \)[/tex].
- The volume [tex]\( V \)[/tex] of a cube is given by the formula:
[tex]\[ V = \text{side}^3 \][/tex]
- We need to find the side length of the cube, which is the cube root of the volume:
[tex]\[ \text{side} = \sqrt[3]{4224} \approx 16 \, \text{cm} \][/tex]
3. Determine the exact volume of the new cube.
- The side length of the cube when rounded to the nearest integer is 16 cm.
- Calculate the volume of this new cube:
[tex]\[ V_{\text{cube}} = 16^3 = 16 \times 16 \times 16 = 4096 \, \text{cm}^3 \][/tex]
4. Find the difference in volumes to understand how much metal needs to be added or removed.
- Compare the volume of the new cube with the volume of the original cuboid:
[tex]\[ \text{Volume difference} = V_{\text{cube}} - V_{\text{cuboid}} \][/tex]
[tex]\[ \text{Volume difference} = 4096 - 4224 = -128 \, \text{cm}^3 \][/tex]
This negative volume difference indicates that the new cube has less volume than the original cuboid. Thus:
- The minimum amount of metal that has to be removed is 128 cm³.
