Introduction / Context:
This question combines the ideas of volume conservation and surface area of a cube. When several solid cubes are melted and recast into a new cube, the total volume of the metal remains the same but the surface area changes. We must compute the combined volume of the original cubes, then find the side length of the new cube, and finally compute its surface area. Such problems are common in aptitude exams to test spatial reasoning and the use of formulas in multiple steps.
Given Data / Assumptions:
- Edge length of first cube = 1 cm.
- Edge length of second cube = 6 cm.
- Edge length of third cube = 8 cm.
- All three cubes are melted and recast into one new cube.
- No loss of material is assumed.
Concept / Approach:
For a cube with side a:
- Volume V = a^3.
- Total surface area S = 6 * a^2.
Since volume is conserved when the cubes are melted and recast, we first find the sum of the volumes of the three cubes. Let that sum be V total. If the side of the new cube is A, then:
A^3 = V total
Then the new surface area S new is:
S new = 6 * A^2
Step-by-Step Solution:
Step 1: Compute volumes of the three cubes.
V1 = 1^3 = 1 cubic centimetre.
V2 = 6^3 = 216 cubic centimetres.
V3 = 8^3 = 512 cubic centimetres.
Step 2: Compute total volume.
V total = 1 + 216 + 512 = 729 cubic centimetres.
Step 3: Find side of new cube.
Let side of new cube be A.
A^3 = 729.
Since 9^3 = 729, A = 9 centimetres.
Step 4: Compute surface area of new cube.
S new = 6 * A^2 = 6 * 9^2 = 6 * 81 = 486 square centimetres.
So, the surface area of the new cube is 486 square centimetres.
Verification / Alternative check:
We can cross check by factoring 729 as 9 * 81 and recognizing that 81 is 9^2, confirming that 729 is indeed 9^3. Also, note that if we had chosen a different side, such as 8 or 10, the volume would have been 512 or 1000 respectively, which are not equal to 729. Thus, 9 cm is the unique correct side length.
Why Other Options Are Wrong:
Option B (586), Option C (686) and Option D (786) all correspond to incorrect side lengths and would not give a cube volume of 729 cubic centimetres. Option E (400) would arise if someone used side 8 or another incorrect value and misapplied the surface area formula. Only 486 corresponds to the correctly computed side length of 9 cm.
Common Pitfalls:
A common mistake is to sum the edges and not the volumes, or to average the sides and treat that as the new side length. Another error is to forget that the new cube side must be the cube root of total volume, not just the square root. Finally, some students incorrectly assume that surface area is conserved, which is not true when shapes change while volume is preserved.
Final Answer:
The surface area of the cube formed is
486 square centimetres.
Discussion & Comments