how many boxes con you find that will hold two times as many cubes as a 2 by 3 by 4 box.
To find the number of boxes that can hold two times as many cubes as a 2 by 3 by 4 box, we first need to calculate the volume of the 2 by 3 by 4 box.
Volume = Length × Width × Height
Volume = 2 × 3 × 4
Volume = 24 cubic units
Now, we need to find boxes that can hold two times as many cubes as this 2 by 3 by 4 box. This means the volume of the new boxes should be double the volume of the 2 by 3 by 4 box.
Double the volume = 2 × 24
Double the volume = 48 cubic units
To find the dimensions of the boxes, we need to consider the factors of 48 that form a valid rectangular box. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
We can arrange these factors into different sets of three numbers (length, width, and height) and calculate their volume. If the volume is 48 cubic units, then that set of dimensions is a valid box.
Let's consider each set of dimensions one by one:
Set 1: 1 × 1 × 48 → Volume = 1 × 1 × 48 = 48 cubic units (valid)
Set 2: 1 × 2 × 24 → Volume = 1 × 2 × 24 = 48 cubic units (valid)
Set 3: 1 × 3 × 16 → Volume = 1 × 3 × 16 = 48 cubic units (valid)
Set 4: 1 × 4 × 12 → Volume = 1 × 4 × 12 = 48 cubic units (valid)
Set 5: 1 × 6 × 8 → Volume = 1 × 6 × 8 = 48 cubic units (valid)
Set 6: 2 × 2 × 12 → Volume = 2 × 2 × 12 = 48 cubic units (valid)
Set 7: 2 × 3 × 8 → Volume = 2 × 3 × 8 = 48 cubic units (valid)
Set 8: 3 × 4 × 4 → Volume = 3 × 4 × 4 = 48 cubic units (valid)
Therefore, there are 8 different sets of dimensions for boxes that can hold two times as many cubes as a 2 by 3 by 4 box.