A cuboid of dimensions 7cm × 5cm × 3cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7cm × 5cm, 5cm × 3cm, 7cm × 3cm respectively. Then the cuboid is cut and separated into various cubes each of side length 1cm. which of the following statements is/are correct?

Correct Option

The cuboid has dimensions 7cm × 5cm × 3cm and is subsequently cut into smaller cubes, each with a side length of 1cm.

The number of small cubes with no paint on any face is determined by considering the internal volume of the cuboid, excluding the outermost layer on all sides. The effective dimensions for these unpainted cubes become (Length - 2) × (Width - 2) × (Height - 2).

Substituting the given dimensions:

• Length: \(7 - 2 = 5\) cm
• Width: \(5 - 2 = 3\) cm
• Height: \(3 - 2 = 1\) cm

Therefore, the number of unpainted cubes is \(5 \times 3 \times 1 = 15\).

Hence, Statement 1, which asserts that there are exactly 15 small cubes with no paint on any face, is correct.

Incorrect Options

Statement 2 claims that there are exactly 6 small cubes with exactly two faces, one painted blue and the other green.

Cubes with exactly two painted faces are located along the edges of the cuboid, excluding the corner cubes. To identify cubes painted blue and green, we must consider the edges where a blue-painted face meets a green-painted face.

• Faces of dimensions 7cm × 3cm are painted blue.
• Faces of dimensions 5cm × 3cm are painted green.

The edges common to both a blue face and a green face are those with a length of 3cm. A cuboid has 4 such edges of identical length.

For an edge of length 'L', the number of cubes with exactly two painted faces is (L-2).

For the 3cm long edges, the number of cubes painted blue and green is \(4 \times (3 - 2) = 4 \times 1 = 4\).

Since the calculation yields 4 cubes, not 6, Statement 2 is incorrect.