Final answer:
The volume of a prism changes to 1/8 of the original volume when all linear dimensions are cut in half, since volume is proportional to the cube of the scaling factor.
Step-by-step explanation:
Understanding the Change in Prism Volume
When all linear dimensions of a prism are cut in half, the volume of the prism changes dramatically. The volume of a prism is calculated by the product of its length, width, and height (V = lwh). If each dimension is reduced by half, the new volume is calculated as V = (l/2)(w/2)(h/2), which simplifies to V = (1/8)lwh. This shows that the new volume is 1/8 of the original volume.
By reducing the dimensions of the prism to half of their original sizes, we are applying the scaling factor of 1/2 to each dimension. Since volume is a measure of three-dimensional space, it's affected by the cube of the scaling factor (1/2)^3, which is 1/8. Therefore, the prism's volume becomes one-eighth of what it was originally when all linear dimensions are cut in half.
The options given (5 in., 4 in., 17 in.) appear to be unrelated to the concept of how volumes change with linear dimensions; instead, they may refer to specific measurements not provided in the context of this question. Thus, understanding the relationship between dimensional changes and volume is key, rather than focusing on these numerical options.