We are given the volume of each wedge [tex]\( V = \frac{1}{2} \pi r^3 \)[/tex] and we need to solve for [tex]\( r \)[/tex].
Let's manipulate this equation step-by-step to isolate [tex]\( r \)[/tex]:
1. Start with the given equation:
[tex]\[
V = \frac{1}{2} \pi r^3
\][/tex]
2. Multiply both sides of the equation by 2 to eliminate the fraction:
[tex]\[
2V = \pi r^3
\][/tex]
3. Divide both sides of the equation by [tex]\(\pi\)[/tex] to isolate [tex]\( r^3 \)[/tex]:
[tex]\[
\frac{2V}{\pi} = r^3
\][/tex]
4. Take the cube root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \sqrt[3]{\frac{2V}{\pi}}
\][/tex]
This is the derived formula for [tex]\( r \)[/tex]. Let’s find which of the multiple-choice options matches this formula:
A. [tex]\( r = \sqrt[3]{\frac{6V}{r}} \)[/tex]
- Incorrect. This expression incorrectly mixes [tex]\( r \)[/tex] within the cube root in a form that doesn't simplify properly.
B. [tex]\( r = \sqrt[3]{6V} \)[/tex]
- Incorrect. This expression does not correctly include [tex]\(\pi\)[/tex] in the denominator.
C. [tex]\( r = \sqrt[3]{\frac{1}{6V}} \)[/tex]
- Incorrect. The form completely conflicts, having terms in the denominator inside the cube root.
D. [tex]\( r = \sqrt[3]{6V(\pi)} \)[/tex]
- Incorrect. This form places [tex]\(\pi\)[/tex] in the numerator improperly and doesn't follow the derived formula.
None of the provided options exactly match [tex]\( r = \sqrt[3]{\frac{2V}{\pi}} \)[/tex], indicating a potential issue with provided choices. If we had to choose the closest correct representation based on a correct derivation, we recognize the correct answer is unfortunately not listed explicitly among the options.
Hence, the exact solution formula here for correctly solving the given equation is:
[tex]\[
r = \sqrt[3]{\frac{2V}{\pi}}
\][/tex]
