Calculus I Optimization

Question

A cylinder is inscribed in a right circular cone of height h  and radius (at the base) equal to r  . What are the dimensions of such a cylinder which has maximum volume?

Answer 1

cylinder inscribed in cone with height h and radius r

what are the cylinder's dimensions that maximize its volume??

cylinder's volume = its height x times base area A = piy^2 where y = radius

V = xpiy^2 find x and y that maximize V

find the relation between x and y, find V(x)

then take the derivative, set it equal to zero and solve for x = volume maximizing cylinder radius

if x = h, y = 0, if y = r, h=0

x = h/2, y= r/2 is midway and maximizes volume

x and y are inversely related

V(x) = xpiy^2

V(h/2) = (h/2)pi(r/2)^2 = (pihr^2)/8 = max Volume