Cylinder inscribed in a cone, hate these problems!?

Question

A cylinder is inscribed in a right circular cone of height 7.5 and radius (at the base) equal to 5. You will be asked to determine the dimensions of such a cylinder which has maximum volume.

a. First, if the radius of such an inscribed cylinder is x, then its height will be (express the cylinder's height as a function of x)

b. Next, give an expression for the volume of the inscribed cylinder in terms of its radius x.

c. Finally, the inscribed cylinder that maximizes its volume will have dimensions:
Height:
Radius:

Answer 1

Use similar triangles to get x/5 =(7.5-h)/7.5. This gives
h = (37.5 -7.5x)/5

V = pix^2h = pix^2(37.5-7.5x)/5

Now find dV/dx and set it equal to 0.
The value of x that gives you a max is your answer.

Answer 2

Okay, letting x be distance from edge of cylinder on bottom to edge of cone at bottom, then by similiar triangles, R/h=x/H. Now area of cylinder is H*pi(R-x)^2. Or H*pi(R-(HR/h))^2= H*pi(R^2-2HR^2/h+H^2R^2/h^2) = pi(HR^2-2H^2R^2/h+H^3R^2/h^2). Now differinate in respect to H and set to 0. pi(R^2-4HR^2/h+3H^2R^2/h^2=0. Or 1-4H/h+3H^2/h^2=0. Using quadratic, H=((4/h+-sqr(16/h^2- (4*3/h^2*1))/(3*2) = =(4/h-2/h)/6 = (2/h)/6 = h/3.