1. If the radius of a cone is \(r\) units, the height is \(h\) units, and the curved surface area is \(S\) square units, then prove that \[ h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r} \]

2. The base radius of a solid right circular cone is \(r\) units, its vertical height is \(h\) units, and its slant height is \(l\) units. The cone’s base is attached to the base of a solid right circular cylinder. If the base radius and height of the cylinder are equal to those of the cone, then the total surface area of the combined solid is \((πrl + 2πrh + 2πr^2)\) square units.

3. Translate: "If a right circular cylinder has lateral surface area \(c\) square units, base radius \(r\) units, and volume \(V\) cubic units, then find the height (value) of the cylinder." Would you like me to solve it too? I’m ready when you are.

4. A right circular cone and a cylinder have equal curved surface areas. If the height and radius of the cone are \(h\) and \(r\), and the height and radius of the cylinder are \(H\) and \(r\), then show that: \[ h^2 = (2H + r)(2H - r) \]

5. If a right circular cone has volume \(V\) cubic units, base area \(A\) square units, and height \(H\) units, then determine the value of \(\cfrac{AH}{3V}\).

6. If the curved surface area of a right circular cylinder is \(c\) square units, the base radius is \(r\) units, and the volume is \(v\) cubic units, then the value of \(\frac{cr}{v}\) is = _____.