on a xyz coordinate system ,xto the right, y into the page, z vertical sketch a right circular cone. the base is a circle,radius 3, centered at the origin and in the xy plane.
The apex is at x=0 y=0 z=5
sketch a line from the apex to the point x=3 y=z=0
the equation of this line is:
z=mx+b at x=0 z=5
5=b
the equation of the line is :
z=mx+5 at z=0 x=3
0=3m+5
m=-5/3
the equation of the line is
z=-5/3 x + 5
At the height z along the z axis, draw a horizontal line, from the axis untill it intersects the line defined above. Draw a circle of this line segments length, about the z axis, with this radius.
x=3[5-z]/5 from above equation
Draw a identical circle about the origin , in the xy plane.
These two circles define the bases of a right circular cylindar inscribed in the cone.
The volume of the cylinder,V, is
V= pi r^2h
where:
r=3[5-z]/5
h=z
V=[9pi/25][5-z]^2 z
V=[9pi/25][25-10z+z^2]z
V=[9pi/25][25z-10z^2+z^3] take derivative with respect to z
dV/dz=[9pi/25][25 -20z+3z^2] set the derivative =0
0=[9pi/25][25-20z+3z^2] but [9pi/25] can't equal 0
0=25-20z+3z^2 factor
0=[5-3z][5-z]
5-3z=0 OR 5-z=0
z=5/3 OR z=5
If z=5 the volume of the cylinder is 0
z=5/3 answer
height = 5/3 feet answer
radius = 3[5-z]/5
radius= 3[5-5/3]/5
radius = [15-5]/5
radius= 2 feet answer
I have assumed that the greatest volume of the inscribed cylinder has its base in the base of the cone. This is not neccessarily true.
Youcan the line segment as defined above as 1/2 the height of a cylinder. The z we defined above as the diameter of the the bases of a right circular cylinder, and see if its maximum volume is greater than the one we just defined
Arthur
