To find the volume of a 3-dimensional figure (in this case, a rectangular prism) with dimensions of 6 cm, 8 cm, and 10 cm, we start by calculating the original volume. The volume \( V \) of a rectangular prism can be computed using the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Substituting the given dimensions:
\[ V = 6 , \text{cm} \times 8 , \text{cm} \times 10 , \text{cm} = 480 , \text{cm}^3 \]
Next, if the figure is dilated using a scale factor of 2, the new dimensions will be:
\[ \text{New length} = 6 , \text{cm} \times 2 = 12 , \text{cm} \] \[ \text{New width} = 8 , \text{cm} \times 2 = 16 , \text{cm} \] \[ \text{New height} = 10 , \text{cm} \times 2 = 20 , \text{cm} \]
Now we calculate the volume of the dilated figure:
\[ V' = \text{New length} \times \text{New width} \times \text{New height} \] \[ V' = 12 , \text{cm} \times 16 , \text{cm} \times 20 , \text{cm} \] \[ V' = 3840 , \text{cm}^3 \]
Alternatively, there is a simpler way to determine the volume of a dilated figure. When a figure is dilated by a scale factor \( k \), the volume scales by a factor of \( k^3 \). In this case, the scale factor is 2:
\[ \text{New Volume} = \text{Original Volume} \times k^3 \] \[ \text{New Volume} = 480 , \text{cm}^3 \times 2^3 \] \[ \text{New Volume} = 480 , \text{cm}^3 \times 8 = 3840 , \text{cm}^3 \]
Thus, the volume of the dilated figure is:
\[ \boxed{3840 , \text{cm}^3} \]
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