Final answer:
The number of marbles a hollow cube with an internal edge of 22 cm can accommodate, given that each marble has a diameter of 0.5 cm and 18% space remains unfilled, is calculated to be 133333. However, this answer does not match any of the provided options.
Step-by-step explanation:
To calculate the number of marbles that a hollow cube can accommodate, we need to first calculate the total volume of the cube and then the volume that is actually usable for the marbles, considering that 18% of it remains unfilled. We have the internal edge of the cube as 22 cm and the diameter of each marble as 0.5 cm.
The volume of the cube (Vcube) is given by the formula Vcube = edge3. Therefore:
Vcube = 22 cm x 22 cm x 22 cm = 10648 cm3.
The volume available for marbles (Vavail) after considering the unfilled space is 82% of Vcube (since 18% is unfilled):
Vavail = 0.82 x 10648 cm3 = 8727.36 cm3.
The volume of a single marble (Vmarble) is given by the formula for the volume of a sphere, which is V = 4/3 x π x r3, where r is the radius of the marble:
Vmarble = 4/3 x π x (0.25 cm)3 = 0.06545 cm3 (approximately).
To find the number of marbles (N) that fit into the available volume, we divide Vavail by Vmarble:
N = Vavail / Vmarble = 8727.36 cm3 / 0.06545 cm3 = 133333.4.
Since we cannot have a fraction of a marble, we round down to the nearest whole number:
N = 133333 marbles.
However, none of the options provided in the question (A. 122444, B. 144244, C. 142442, D. 142244) matches our calculation. If the student made an error in transcribing the question, they should check the original problem to ensure the correct data and options were presented. If not, it is possible there was a misunderstanding or typo in the question provided.