Let's solve this problem step-by-step:
1. Calculate the original volume of the prism:
- The original base area ([tex]\(A\)[/tex]) of the triangular prism is [tex]\(20 \text{ cm}^2\)[/tex].
- The original height ([tex]\(h\)[/tex]) of the triangular prism is [tex]\(13 \text{ cm}\)[/tex].
- The volume ([tex]\(V\)[/tex]) of a prism is given by the formula:
[tex]\[
V = A \times h
\][/tex]
- Substituting the given values:
[tex]\[
V = 20 \text{ cm}^2 \times 13 \text{ cm} = 260 \text{ cm}^3
\][/tex]
2. Calculate the new volume after dilation:
- The scale factor ([tex]\(k\)[/tex]) for the dilation is [tex]\(0.5\)[/tex].
- When a geometric shape is dilated, each linear dimension (length, width, height) is multiplied by the scale factor. Since the volume depends on all three dimensions (in three-dimensional space), the volume will be affected by the cube of the scale factor:
[tex]\[
\text{New volume} = \text{Original volume} \times k^3
\][/tex]
- Substituting the values:
[tex]\[
\text{New volume} = 260 \text{ cm}^3 \times (0.5)^3
\][/tex]
- Calculate [tex]\( (0.5)^3 \)[/tex]:
[tex]\[
(0.5)^3 = 0.5 \times 0.5 \times 0.5 = 0.125
\][/tex]
- Now, multiply the original volume by this factor:
[tex]\[
\text{New volume} = 260 \text{ cm}^3 \times 0.125 = 32.5 \text{ cm}^3
\][/tex]
Thus, the new volume of the prism after dilation is [tex]\(32.5 \text{ cm}^3\)[/tex].
None of the options provided (260 cm³, 130 cm³, 65 cm³, 325 cm³) are correct based on this detailed solution. The correct volume after dilation is indeed [tex]\(32.5 \text{ cm}^3\)[/tex].
