Determine the formula for the Volume of the cylinder contained within a 1 m radius sphere relative to the radius of the cylinder.
Differentiate to establish critical points.
Volume of cylinder (V) = pi r^2h = 2 pi r^2 s
where s is the semi-height.
r^2 + s^2 = 1 rarr s = sqrt(1-r^2)
so
V(r) = 2pi r^2 * (1-r)^(1/2)
(d V(r))/(dr) = 2pi ( (d r^2)/(dr)*(1-r^2)^(1/2) + r^2 (d (1 - r^2)^(1/2) )/(dr) )
Since
(d r^2)/(dr) = 2r
and
(d (1-r^2)^(1/2))/(dr) = (d (1-r^2)^(1/2))/(d(1-r^2)) * (d (1-r^2))/(dr)
= (- r) (1-r^2)^(- 1/2)
(d V(r))/(dr) = 2pi (2r (1-r^2)^(1/2) + r^2( (-r)/(sqrt(1-r^2))))
= 2 pi ( 2rsqrt(1-r^2) - (r^3)/(sqrt(1-r^2)))
Set (d V(r))/(dr) = 0 for critical points
2 pi ( 2rsqrt(1-r^2) - (r^3)/(sqrt(1-r^2)))= 0
If r != 0 we can divide by 2 pi r
2 sqrt(1-r^2) - (r^2)/(sqrt(1-r^2)) = 0
If r != 1 we can multiply by sqrt(1-r^2)
2 (1-r^2) - r^2 = 0
2 - 3r^2 = 0
r = sqrt(2/3)
Substituting we can find
s = (1 - 2/3)^(1/2) = sqrt(1/3)
and
h = 2 sqrt(1/3)
Note the extraneous possibilities r=0 and r=1 give the minimum cylinder size (which should be obvious from observation).
