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Difficulty: 4/102024 IOQM 2024 (Q7)

Determine the sum of all possible surface areas of cube two of whose vertices are (1, 2, 0) and (3, 3, 2).

Guide / Hint

Hint 1: Start by analyzing the initial conditions and setting up the basic equations. SA = 6 \times  4 + 1 + 4 .

Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. = 54 sq.unit2.

Hint 3: Proceed with the final algebraic steps to solve the system. 3 9.

Solution

Step 1: SA = 6 \times  4 + 1 + 4 

Step 2: = 54 sq.unit2

Step 3: 3 9

Step 4: a= SA = 6 = 27 sq unit

Step 5: 2 2

Step 6: SA = 6a2 = 6 × 3 = 18 sq.unit2

Step 7: => Possible SA = 18 + 27 + 54 = 99

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