. 5 Change the subject of the formula to \( k: G=\frac{1}{3} s^{3}+\frac{1}{2} k^{2}-3 \)

To change the subject of the formula to \( k \), we start from the equation \( G = \frac{1}{3} s^{3} + \frac{1}{2} k^{2} - 3 \). First, we add 3 to both sides to isolate the term with \( k \): \[ G + 3 = \frac{1}{3} s^{3} + \frac{1}{2} k^{2} \] Next, we subtract \( \frac{1}{3} s^{3} \) from both sides: \[ G + 3 - \frac{1}{3} s^{3} = \frac{1}{2} k^{2} \] To get rid of the fraction, multiply both sides by 2: \[ 2(G + 3 - \frac{1}{3} s^{3}) = k^{2} \] Finally, take the square root of both sides to solve for \( k \): \[ k = \pm \sqrt{2(G + 3 - \frac{1}{3} s^{3})} \] And there you have \( k \) expressed in terms of \( G \) and \( s \)!