A right circular cylinder is inscribed in a cone with a heigh h and base radius r. Find the largest possible volume of such a cylinder.

Answer 1

Let R and H radius and height of inscribed cylinder. Then from cimilyarity of triangles wehave: H/h = (r - R)/r.

From here H = h(r - R)/r.

V = 1/3πR²H = 1/3πR²h(r - R)/r; We have now volume as function of one variable R;

dv/dR = 2/3π·Rh(r - R)/r - 1/3πR²h/r = 0; 2R( r - R) - R² = 0; R = 0 or 2r - 2R - R = 0; R = 2/3r

d²V/dR² = 2/3πh(r - 2R)/r - 2/3πRh/r = 2/3πh(r - 4R)/r;

Now we see that d²V/dR² < 0 at R = 2/3r we have maximum volume of inscribed cone.

H = h(r - 2/3r)/r = h/3.

So max V = 1/3π(2/3r)²h/3 = 4πr²h/81