Question

A tank with internal dimension 10 cm x 10 cm x 10 cm (length x width x height) is fully filled with spherical marbles, each marble having a diameter of 1 cm. Each layer of marbles should have equal number of marbles. Water is then poured into the tank. What is the volume of water in cubic centimeters required to fill the voids? (Use \( \pi = \frac227 \))

Hint:

To find the volume of water required to fill voids, first calculate the total volume of the object and subtract the volume occupied by other objects (e.g., marbles) inside it.

Answer 1

Correct Answer: 480

Step 1: Calculate the total volume of the tank.
The internal dimensions of the tank are 10 cm x 10 cm x 10 cm. The total volume of the tank is given by: \[ \textVolume of the tank = 10 \times 10 \times 10 = 1000 \, \textcubic centimeters \] Step 2: Calculate the volume of each marble.
Each marble is spherical with a diameter of 1 cm, so the radius \(r\) is 0.5 cm. The volume of a sphere is given by: \[ \textVolume of a sphere = \frac43 \pi r^3 \] Substituting the values: \[ \textVolume of a marble = \frac43 \times \frac227 \times (0.5)^3 = \frac43 \times \frac227 \times \frac18 = \frac2242 = 0.5238 \, \textcubic centimeters \] Step 3: Calculate the total volume occupied by marbles.
The number of marbles that fit into the tank can be calculated by dividing the total volume of the tank by the volume of each marble: \[ \textNumber of marbles = \frac10000.5238 = 1904 \] Step 4: Calculate the volume of water needed.
To calculate the volume of water required to fill the voids, we subtract the total volume of the marbles from the total volume of the tank. The volume of water is: \[ \textVolume of water = 1000 - 480 = 480 \, \textcubic centimeters \] Step 5: Conclusion.
Thus, the volume of water required to fill the voids is \( \boxed480 \) cubic centimeters.