Which of the following is the total number of whole boxes that measure 2 ft * 2 ft * 2 ft that can be stored in a room that measures 9 ft * 9 ft * 9 ft, if the size of the boxes cannot be altered?
A.
125
B.
64
C.
92
D.
18
The correct answer and Explanation is :
To solve this problem, let’s break it down step by step:
Step 1: Volume of the Room
The dimensions of the room are 9 ft * 9 ft * 9 ft. To find the volume of the room, we use the formula for the volume of a cube:
[
\text{Volume of room} = \text{side}^3 = 9^3 = 729 \text{ cubic feet.}
]
Step 2: Volume of a Single Box
The dimensions of each box are 2 ft * 2 ft * 2 ft. Similarly, we find the volume of one box:
[
\text{Volume of one box} = \text{side}^3 = 2^3 = 8 \text{ cubic feet.}
]
Step 3: Number of Boxes by Volume (Preliminary Calculation)
If we divide the volume of the room by the volume of one box, we get the theoretical maximum number of boxes that could fit into the room (ignoring practical considerations):
[
\text{Number of boxes} = \frac{\text{Volume of room}}{\text{Volume of one box}} = \frac{729}{8} = 91.125.
]
Since we cannot have a fraction of a box, we round down to 91 boxes. But this is just a preliminary calculation. We need to consider the spatial arrangement next.
Step 4: Number of Boxes by Spatial Arrangement
Since both the room and the boxes are cubes, the number of whole boxes that can be stored depends on how many fit along each dimension. We find how many boxes fit along each side of the room:
[
\frac{\text{Room side}}{\text{Box side}} = \frac{9}{2} = 4.5.
]
Again, we cannot have a fraction of a box along any side, so only 4 boxes can fit along each dimension (length, width, and height).
Step 5: Total Number of Boxes
Now, to find the total number of boxes that fit, we multiply the number of boxes that fit along each dimension:
[
\text{Total boxes} = 4 \times 4 \times 4 = 64.
]
Conclusion:
The total number of whole boxes that can fit in the room is 64.
Therefore, the correct answer is B. 64.