A hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm. If \(\dfrac{1}{8}\) of the space remains unfilled, then the number of marbles the cube can accommodate is
142296
142396
142496
142596
Step 1: Find the volume of the cube.
The internal edge of the cube = 22 cm.
Volume of cube = \(a^3 = 22^3 = 10648\,\text{cm}^3\).
Step 2: Account for the empty space.
Only \(\tfrac{7}{8}\) of the cube is filled with marbles.
Filled volume = \(\tfrac{7}{8} \times 10648 = 9317\,\text{cm}^3\).
Step 3: Find the volume of one marble.
Diameter of one marble = 0.5 cm, so radius = 0.25 cm.
Volume of one sphere = \(\tfrac{4}{3}\pi r^3 = \tfrac{4}{3}\pi (0.25)^3\).
= \(\tfrac{4}{3}\pi (0.015625) = 0.020833\pi \approx 0.06545\,\text{cm}^3\).
Step 4: Find the number of marbles.
Total filled volume ÷ Volume of one marble = \(\tfrac{9317}{0.06545}\).
≈ 142396 marbles.
Final Answer: Option B (142396)