Maxima And Minima

Step 1: Consider the geometry. Let a right circular cone of height and base radius be inscribed in a sphere of radius . - Vertex of cone at top of sphere. - Base of cone inside sphere. By Pythagoras (in right-angled triangle from sphere center to cone base):

Step 2: Write volume of cone.

Step 3: Maximize volume. Differentiate: Set : So, or . We take .

Step 4: Compute . So, .

Step 5: Maximum volume of cone.

Step 6: Volume of sphere.

Step 7: Ratio.

Final Answer: