The volume of a prism is given by the formula V = (1/3) * A * h, where A is the base area and h is the height.
Let's denote the base area of the pentagonal prism A as A_A and the height as h_A. Similarly, let's denote the base area of the pentagonal prism B as A_B and the height as h_B.
Since pentagonal prism B is the image of pentagonal prism A after dilation by a scale factor of 1/4, we know that the base area of B is (1/4)^2 = 1/16 times the base area of A, and the height of B is 1/4 times the height of A.
Therefore, A_B = (1/16) * A_A and h_B = (1/4) * h_A.
Given that the volume of pentagonal prism A is 128 in^3, we have:
V_A = (1/3) * A_A * h_A = 128
Substituting for A_A and h_A in terms of A_B and h_B:
V_A = (1/3) * (16*A_B) * (4*h_B) = 128
V_A = (64/3) * A_B * h_B = 128
A_B * h_B = (3/64) * 128
A_B * h_B = 6
Therefore, the volume of pentagonal prism B, V_B, is:
V_B = (1/3) * A_B * h_B
V_B = (1/3) * 6
V_B = 2 in^3
Therefore, the volume of pentagonal prism B is 2 cubic inches.
Pentagonal prism B is the image of pentagonal prism A after dilation by a scale factor of one quarter
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. If the volume of pentagonal prism A is 128 incubed
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, find the volume of pentagonal prism B, the image.
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