Question 1191566
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Here is a 2-dimensional sketch of a right circular cylinder inscribed in a right circular cone.<br>
{{{drawing(400,400,-8,8,-2,14
,line(-5,0,5,0),line(-5,0,0,12),line(5,0,0,12),line(0,0,0,12),line(-10/3,4,10/3,4)
,line(-10/3,0,-10/3,4),line(10/3,0,10/3,4)
,locate(0,0,A),locate(.2,3.8,B),locate(0,13,C),locate(5,0,D),locate(3.6,4.2,E)
)}}}<br>
We are given AD = 5 and CD = 13, so we can conclude AC = 12.<br>
Let x be the radius of the cylinder, BE.  Similar triangles ACD and BCE give us BC = (12/5)x = 2.4x.  The height of the cylinder is then 12-2.4x.<br>
Express the volume of the cylinder in terms of x and maximize the volume by finding where the derivative of the volume function is 0.<br>
{{{V=(pi)(r^2)(h)=(pi)(x^2)(12-2.4x)=pi(12x^2-2.4x^3)}}}<br>
{{{dV/dx = pi(24x-7.2x^2) = (pi)(24x(1-.3x))}}}<br>
{{{.3x = 1}}}
{{{x = 1/.3 = 10/3}}}<br>
The volume is maximum when the radius of the cylinder is 10/3.<br>
That makes the height {{{12-2.4(10/3)=12-8=4}}}<br>
ANSWER: radius 10/3; height 4<br>