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Alright listen up, I'll try to explain it for you.
Now, let k = radius of a right circular cylinder inscribed within a sphere of radius r where h is the height of the cylinder.
Then the volume of the cylinder, (I’ll use Vc), is Vc = pi*k2*h.
Note that the distance from the center of the cylinder (same as center of sphere) to a point on the rim of the cylinder (also on sphere) is r. Note also that the (shortest) distance from the center of cylinder to the side of the cylinder is k. Note even further, that the distance from the point on the side (just mentioned) to a point on the rim if cylinder is h/2. A diagram can help you see that the points mentioned above form a right triangle. Label the central angle of this triangle as T.
Then sin(T) = h/(2r) and cos(T)=k/r
this gives h = 2r*sin(T) and k = r*cos(T)
So, we can rewrite the volume of the (inscribed) cylinder as
Vc = pi*r2cos2(T)*2r*sin(T)
giving Vc = 2pi*r3cos2(T)sin(T)
...Maximize Vc.
differentiate with respect to T (remember that r is considered constant here).
Vc’ = 2pi*r3cos(T)*[cos2(T) - 2sin2(T)] ...this is after some simplifying/rearranging.
Keep in mind that T must be between 0 and pi/2 exclusive (refer to your diagram)...
critical values will be when Vc’ = 0.
this should lead you to cos2(T) - 2sin2(T) {since the other factor will have no critical values in the desired range}
further, this gives tan(T) = sqrt(2)/2 {remember there are range restrictions which allow us to disregard other possibilities}
This gives sin(T) = sqrt(3)/3 and cos(T) = sqrt(6)/3.
Substituting the values into our volume equation for a cylinder gives
Vc = 2pi*r3cos2(T)sin(T)
= (4/9)sqrt(3)pi*r3 as the maximum volume of an inscribed cylinder
phew... {and I even left out some of the stuff}
so, the ratio of volume of sphere to maximum volume of inscribed cylinder is
Vs to Vc
or [(4/3)pi*r3] / [(4/9)sqrt(3)pi*r3]
which yields sqrt(3) to 1 after simplifying.
{...so your given answer was correct}
sure hope you followed all that ‘cause i’m tired of typing
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