To find the volume of a rectangular prism, you use the formula:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]
Given the dimensions:
- Height (tall) = 5
- Length (long) = 8
- Width = 4
First, calculate the volume of the original rectangular prism:
\[ \text{Volume} = 8 \times 4 \times 5 \]
Calculating this step-by-step:
\[ 8 \times 4 = 32 \]
\[ 32 \times 5 = 160 \]
So, the volume of the original rectangular prism is 160 cubic units.
Now, if the prism is dilated using a scale factor of 12, all dimensions (height, length, and width) are multiplied by 12. The new dimensions would be:
- New Height = \( 5 \times 12 = 60 \)
- New Length = \( 8 \times 12 = 96 \)
- New Width = \( 4 \times 12 = 48 \)
Now, we calculate the volume of the dilated rectangular prism:
\[ \text{Volume} = \text{new length} \times \text{new width} \times \text{new height} \]
\[ \text{Volume} = 96 \times 48 \times 60 \]
To simplify the calculation, we first find \( 96 \times 48 \):
\[ 96 \times 48 = 4608 \]
Now multiply this by 60:
\[ 4608 \times 60 = 276480 \]
Thus, the volume of the dilated rectangular prism is:
\[ \text{Volume} = 276480 \text{ cubic units} \]
Alternatively, you can also find the volume by recognizing that if a prism is dilated by a scale factor of \( k \), the new volume is \( k^3 \) times the original volume. Here, \( k = 12 \), so:
\[ \text{New Volume} = 12^3 \times \text{Original Volume} = 1728 \times 160 \]
Calculating this gives:
\[ 1728 \times 160 = 276480 \]
Thus, the final answer for the volume of the dilated rectangular prism is:
\[ \text{Volume} = 276480 \text{ cubic units} \]
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