Let's analyze how the volume and surface area of a sphere change when its radius is divided by 2.
1. Volume Change:
The volume [tex]\( V \)[/tex] of a sphere is given by the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
If the radius [tex]\( r \)[/tex] is divided by 2, the new radius becomes [tex]\( \frac{r}{2} \)[/tex]. Let's substitute this new radius into the volume formula:
[tex]\[
V_{\text{new}} = \frac{4}{3} \pi \left(\frac{r}{2}\right)^3 = \frac{4}{3} \pi \frac{r^3}{2^3} = \frac{4}{3} \pi \frac{r^3}{8} = \frac{1}{8} \left(\frac{4}{3} \pi r^3\right) = \frac{1}{8} V
\][/tex]
Therefore, the new volume is [tex]\(\frac{1}{8}\)[/tex] of the original volume. So, the statement "New volume will be 1/8 as much as the original volume" is correct, and the statement "Volume will be divided by 8" is also correct.
2. Surface Area Change:
The surface area [tex]\( S \)[/tex] of a sphere is given by the formula:
[tex]\[
S = 4 \pi r^2
\][/tex]
If the radius [tex]\( r \)[/tex] is divided by 2, the new radius becomes [tex]\( \frac{r}{2} \)[/tex]. Let's substitute this new radius into the surface area formula:
[tex]\[
S_{\text{new}} = 4 \pi \left(\frac{r}{2}\right)^2 = 4 \pi \frac{r^2}{2^2} = 4 \pi \frac{r^2}{4} = \pi r^2 = \frac{1}{4} \left(4 \pi r^2\right) = \frac{1}{4} S
\][/tex]
Therefore, the new surface area is [tex]\(\frac{1}{4}\)[/tex] of the original surface area. So, the statement "New S.A. will be 1/4 of the original surface area" is correct, and the statement "S.A. will be divided by 4" is also correct.
The incorrect statement is "S.A. will also be divided by 2". As we determined earlier, the surface area is divided by 4, not 2.
Summary of the correct statements:
- New volume will be 1/8 as much as the original volume.
- S.A. will be divided by 4.
- New S.A. will be 1/4 of the original surface area.
- Volume will be divided by 8.
