Calculus
posted by Mishaka
Given a right circular cone, you put an upsidedown cone inside it so that its vertex is at the center of the base of the larger cone, and its base is parallel to the base of the larger cone. If you choose the upsidedown cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let H and R be the height and radius of the large cone, let h and r be the height and radius of the small cone. Use similar triangles to get an equation relating h and r. The formula for the volume of a cone is V = 1/2pir^2h.)

Mishaka
Pretty sure I figured it out, 4/27. I found this by simplifying:
((1/3pi (h  2/3 h))(4/9 r^2)) / (1/3 pi r^2 h) 
JWAL
How did you get this? Please help I am stuck.

Steve
Draw the cones. If the large cone has height H and radius R, the small cone has height h and radius r, so that
r/R = 1  h/H
The ratio of the two volumes is v/V = r^2h/R^2H = (r/R)^2 h/H
if we maximize that ratio as a function of u = h/H, we get
f(u) = (1u)^2 * u
= (u  2u^2 + u^3)
f'(u) = (14u+3u^2)
= (h1)(3h1)
Clearly, when u=1, f(u) = 0 a minimum
when u = 1/3, f(h) = 4/9*1/3 = 4/27
so, h/H = 1/3 for max volume ratio 
:)
Okay, letting x be height and b radius of smaller,
then x/b=h/r, and
volumeof smaller is 1/3pi(hx)b^2, or
1/3pi(hx)(x^2r^2/h^2=
1/3pi((x^2r^2/h)(x^3r^2/h^2).
Now differinate with respect to x,
and set to 0, so
((2xr^2/h)(3x^2r^2/h^2)=0,
so you get x=2/3h.
So, fraction is ((1/3pi(h2/3h)(4/9r^2)/(1/3pir^2h)=
4/27
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