A right circular cylinder is inscribed in a right circular cone with altitude 7 and radius of the base 6. What is the greatest possible volume of the cylinder?

Question

A right circular cylinder is inscribed in a right circular cone with altitude 7 and radius of the base 6.What is the greatest possible volume of the cylinder? 

Answer 1

For every unit rise in h, we lose 6/7 unit in radius: r = 6 - 6/7 h

Seeking the equal tradeoff in pct chg V/ pct chg h versus pct chg dV/ pct chg r.

V rt-cyl = pi h rr

dV/dh * h/V = -dV/dr * r/V ... chain rule ... dV/dr * dr/dh * r/V

h dV/dh = -r dV/dr * -6/7 h

h (pi rr) = 6 rh/7 (2pi rh) =12pi/7 rrhh

h = 7 / 12pi

r = 6 - 1 / 2pi ... a pretty flat disc ... makes sense since Vol = f [r-sq] :)