Let's break down how the changes in the dimensions of a rectangular prism affect its surface area.
1. Original Surface Area Formula:
For a rectangular prism with length [tex]\( l \)[/tex], width [tex]\( w \)[/tex], and height [tex]\( h \)[/tex], the surface area [tex]\( S_A \)[/tex] is calculated as:
[tex]\[
S_A = 2lw + 2lh + 2wh
\][/tex]
2. Changes in Dimensions:
- The length [tex]\( l \)[/tex] is shrunk by a factor of two-thirds, so the new length is [tex]\( \frac{2}{3}l \)[/tex].
- The width [tex]\( w \)[/tex] is quadrupled, so the new width is [tex]\( 4w \)[/tex].
- The height [tex]\( h \)[/tex] is cut in half, so the new height is [tex]\( \frac{1}{2}h \)[/tex].
3. New Surface Area Calculation:
Substituting the new dimensions into the surface area formula:
[tex]\[
S_A = 2 \left( \frac{2}{3}l \right) (4w) + 2 \left( \frac{2}{3}l \right) \left( \frac{1}{2}h \right) + 2 (4w) \left( \frac{1}{2}h \right)
\][/tex]
Simplifying each term:
- For the first term:
[tex]\[
2 \left( \frac{2}{3}l \right) (4w) = 2 \left( \frac{8}{3}lw \right) = \frac{16}{3}lw
\][/tex]
- For the second term:
[tex]\[
2 \left( \frac{2}{3}l \right) \left( \frac{1}{2}h \right) = 2 \left( \frac{2}{6}lh \right) = \left( \frac{2}{3}lh \right)
\][/tex]
- For the third term:
[tex]\[
2 (4w) \left( \frac{1}{2}h \right) = 2 (2wh) = 4wh
\][/tex]
Combining all the simplified terms, the new surface area formula is:
[tex]\[
S_A = \frac{16}{3}lw + \frac{2}{3}lh + 4wh
\][/tex]
4. Matching with the Given Choices:
Compare the derived formula with the given options.
The correct formula is:
[tex]\[
S_A = \frac{16}{3}lw + \frac{2}{3}lh + 4wh
\][/tex]
Thus, the best answer from the provided choices is:
[tex]\[
\boxed{C. \ S A = \frac{16}{3}lw + \frac{2}{3}lh + 4wh}
\][/tex]
