To determine the volume of a rectangular prism after dilation, you need to know the original dimensions and how dilation affects volume.
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Understanding Dilation: When you dilate a three-dimensional shape by a scale factor \( k \), each dimension (length, width, height) of the shape is multiplied by \( k \). Therefore, the formula for the new volume \( V' \) after dilation is: \[ V' = k^3 \times V \] where \( V \) is the original volume of the prism.
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Given Scale Factor: You mentioned a scale factor of 4. Therefore, \( k = 4 \), and we need to calculate \( k^3 \): \[ k^3 = 4^3 = 64 \]
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Original Volume: Next, you would need the original volume \( V \) of the prism. If the volume after dilation is given as \( 1,536 , \text{m}^3 \), we can set up the equation: \[ V' = 64 \times V \] So, \[ 1,536 = 64 \times V \]
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Solve for \( V \): To find the original volume \( V \): \[ V = \frac{1,536}{64} \] Calculating this gives: \[ V = 24 , \text{m}^3 \]
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Conclusion: Therefore, if the original volume of the rectangular prism is \( 24 , \text{m}^3 \), and when dilated by a scale factor of 4, the volume becomes \( 1,536 , \text{m}^3 \) as you mentioned.
In summary, if the volume of the rectangular prism after dilation is \( 1,536 , \text{m}^3 \), it indicates that the original volume was \( 24 , \text{m}^3 \) before being dilated by a scale factor of \( 4 \).
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