Explanation:

To find the length of the cardboard box:

- It's given that length : breadth : height = 5 : 3 : 7.

- Let the common ratio be x.

- Therefore, length = 5x, breadth = 3x, and height = 7x.

- The total surface area of a closed box = 2(lb + bh + hl).

- Given surface area = 1278 cm².

- Substitute values: 2(5x·3x + 3x·7x + 7x·5x) = 1278.

- This simplifies to 2(15x² + 21x² + 35x²) = 1278.

- Solving: 2(71x²) = 1278 -> 142x² = 1278.

- Divide both sides by 142: x² = 9.

- x = 3 (since length cannot be negative).

- Therefore, length = 5x = 5·3 = 15 cm.

- Option 2: 15 cm is the correct answer.