Maximize Volume of Right Circular Cone with Constant Slant Edge

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To maximize the volume of a right circular cone with a constant slant edge of 6 cm, the relationship between the height and radius must be established. The volume V of the cone can be expressed in terms of the slant edge length x, leading to the formula V = (1/3)πr²h. By applying calculus, the maximum volume occurs when the height is determined to be 4.8 cm, resulting in a maximum volume of approximately 48π cm³. This analysis highlights the geometric relationships and optimization techniques involved in maximizing the volume of a right circular cone.
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1. The slant edge of a right circular cone is 6 cm in length. Find the height of the cone when the volume is a maximum.

2. Find the maximum volume of a right circular cone whose slant edge has a constant length measure a.
 
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I would add,
3. Express the volume V(x) of a right circular cone in terms of the length x of its slant edge.

Then solve 3, then 2, then 1.
 

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