Questions: Solve for (h). Express your answer in simplest and exact form. (L = pi h^2 + pi h s) (h = )

Solve for \( h \) in the equation \( L = \pi h^{2} + \pi h s \).

Rearranging the equation.

We start with the equation \( L = \pi h^{2} + \pi h s \) and rearrange it to form a standard quadratic equation: \[ \pi h^{2} + \pi h s - L = 0. \]

Applying the quadratic formula.

Using the quadratic formula \( h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = \pi \), \( b = \pi s \), and \( c = -L \), we find: \[ h = \frac{-\pi s \pm \sqrt{(\pi s)^{2} + 4\pi L}}{2\pi}. \] This simplifies to: \[ h = \frac{-s}{2} \pm \frac{\sqrt{4L + \pi s^{2}}}{2\sqrt{\pi}}. \]

The solutions for \( h \) are: \[ h = \frac{-s}{2} - \frac{\sqrt{4L + \pi s^{2}}}{2\sqrt{\pi}} \quad \text{and} \quad h = \frac{-s}{2} + \frac{\sqrt{4L + \pi s^{2}}}{2\sqrt{\pi}}. \]

The solutions for \( h \) are: \[ \boxed{h = \frac{-s}{2} - \frac{\sqrt{4L + \pi s^{2}}}{2\sqrt{\pi}}} \] and \[ \boxed{h = \frac{-s}{2} + \frac{\sqrt{4L + \pi s^{2}}}{2\sqrt{\pi}}}. \]