To find the volume of a sphere in terms of [tex]\(x\)[/tex], given that the radius [tex]\(r\)[/tex] is [tex]\(\frac{3}{2} x\)[/tex], we follow these steps:
1. Recall the formula for the volume of a sphere:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
2. Substitute the given radius [tex]\( r = \frac{3}{2} x \)[/tex] into the volume formula:
[tex]\[
V = \frac{4}{3} \pi \left(\frac{3}{2} x\right)^3
\][/tex]
3. Simplify the expression [tex]\(\left(\frac{3}{2} x\right)^3\)[/tex]:
[tex]\[
\left(\frac{3}{2} x\right)^3 = \left(\frac{3}{2}\right)^3 x^3 = \frac{3^3}{2^3} x^3 = \frac{27}{8} x^3
\][/tex]
4. Substitute [tex]\(\frac{27}{8} x^3\)[/tex] back into the volume formula:
[tex]\[
V = \frac{4}{3} \pi \left(\frac{27}{8} x^3\right)
\][/tex]
5. Simplify the multiplication:
[tex]\[
V = \frac{4}{3} \pi \cdot \frac{27}{8} x^3 = \frac{4 \cdot 27}{3 \cdot 8} \pi x^3 = \frac{108}{24} \pi x^3
\][/tex]
6. Reduce the fraction:
[tex]\[
\frac{108}{24} = 4.5
\][/tex]
7. Thus, the volume of the sphere in terms of [tex]\(x\)[/tex] is:
[tex]\[
V = 4.5 \pi x^3
\][/tex]
Therefore, the correct option is not listed among the choices provided [tex]\(2 \pi x^3\)[/tex], [tex]\(\frac{9}{2} \pi x^3\)[/tex], [tex]\(\frac{27}{8} \pi x^3\)[/tex], and [tex]\(\frac{4}{3} \pi x^3\)[/tex], but rather the correct expression for the volume in terms of [tex]\(x\)[/tex] is:
[tex]\[4.5 \pi x^3\][/tex]
