To solve the equation [tex]\( N = \frac{9}{4} \pi s^4 \)[/tex] for [tex]\( s \)[/tex], we need to isolate [tex]\( s \)[/tex] on one side of the equation. Here's a detailed, step-by-step solution:
1. Start with the given equation:
[tex]\[
N = \frac{9}{4} \pi s^4
\][/tex]
2. To isolate [tex]\( s^4 \)[/tex], divide both sides of the equation by [tex]\( \frac{9}{4} \pi \)[/tex]:
[tex]\[
\frac{N}{\frac{9}{4} \pi} = s^4
\][/tex]
Simplify the fraction on the left side:
[tex]\[
s^4 = \frac{4N}{9\pi}
\][/tex]
3. Now, take the fourth root of both sides to solve for [tex]\( s \)[/tex]:
[tex]\[
s = \pm \left( \frac{4N}{9\pi} \right)^{1/4}
\][/tex]
Note that taking the fourth root yields two pairs of real and imaginary roots:
4. The full set of solutions for [tex]\( s \)[/tex] includes both real and imaginary components. Therefore, [tex]\( s \)[/tex] can be:
- The positive real fourth root:
[tex]\[
s = 0.613291438903102 * N^{1/4}
\][/tex]
- The negative real fourth root:
[tex]\[
s = -0.613291438903102 * N^{1/4}
\][/tex]
- The positive imaginary fourth root:
[tex]\[
s = 0.613291438903102i * N^{1/4}
\][/tex]
- The negative imaginary fourth root:
[tex]\[
s = -0.613291438903102i * N^{1/4}
\][/tex]
where [tex]\( 0.613291438903102 \)[/tex] is the numerical value derived from the fourth root of [tex]\(\frac{4}{9\pi}\)[/tex].
In summary, the solutions for [tex]\( s \)[/tex] are:
[tex]\[
s = \pm 0.613291438903102 \cdot N^{1/4}, \quad s = \pm 0.613291438903102i \cdot N^{1/4}
\][/tex]
