Answer :
The volume of pentagonal prism B, the image, is 448 cubic in.
The volume of a prism is directly proportional to the cube of the scale factor. Therefore, if prism B is the image of prism A after dilation by a scale factor of 2, the volume of prism B will be 2³ = 8 times the volume of prism A.
Given that the volume of prism A is 56 in³. Then the volume of prism B is calculated as,
=> Volume of prism A × (Scale factor)³
=> 56 in³ × 8
=> 448 in³
More about the dilation link is given below.
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The volume of pentagonal prism B is 448 [tex]in^{3}[/tex].
Firstly, when a three-dimensional shape is dilated by a scale factor of k, the volume of the resulting shape changes by a factor of [tex]k^{3}[/tex]. This is because volume is a three-dimensional measure.
Given:
The scale factor, k=2
The volume of pentagonal prism A, [tex]V_{A}[/tex]=56 [tex]in^{3}[/tex]
- Using the formula for the volume change:
[tex]V_{B}[/tex] = [tex]k^{3}[/tex] × [tex]V_{A}[/tex]
Where [tex]V_{B}[/tex] is the volume of prism B.
- Substitute the values into the formula:
[tex]V_{B}[/tex]=[tex]2^{3}[/tex]×56
[tex]V_{B}[/tex]=8×56
[tex]V_{B}[/tex]=448 [tex]in^{3}[/tex]
