(calculus)Find the maximum volume of a cylinder that will fit inside a sphere of radius 1.

Any light on the matter appreciated.

V(cylinder)=pi(r^2)h

And h^2+(2r)^2=4, from the Radius (sphere)=1, diameter = 2

4r^2=4-h^2
r^2=(4-h^2)/4

V= pi(4-h^2)h/4=pi(h)-pi(h^3)/4

V'= pi- 3pi(h^2)/4=0

3pi(h^2)=4pi

h^2= 4/3

h=2/sqr3

r^2=(4-h^2)/4= (4-4/3)/4= 2/3

V= pi(2/3)(2/sqr3)= 4(pi)sqr(3)/9


Hoping this helps!

Think about the cylinder inside of the sphere, and imagine slicing the whole thing in half along a great circle (from north pole to south pole) through the top and bottom of the cylinder.

If you look at the cross section, you'd see a rectangle inside of a circle. The rectangle would have height h and length 2r where h and r are the height and radius of the cylinder. If you draw a line from corner to corner of this rectangle, this diagonal would be length 2---it is a diameter of the sphere.

Using the Pythagorean theorem

h² + (2r)² = 2² ==> r² = 1 - h²/4.

The volume of the cylinder is V = π/3 r² h = (π/3)(h - h^3/4). This gives the volume as a function of the height alone.

Just find dV/dh. Look for the critical number(s) and make sure you're finding a maximum.

I'll let you fill in the details. I get the maximum volume to be 4π/(9√3).