A cylinder with open top with radius r and height h has surface area 8 cm2 . Find the largest possible volume of such a cylinder?

Question

A cylinder with open top with radius r and height h has surface area 8 cm^2 . Find the largest possible volume of such a cylinder?

Answer 1

The surface area of the cylinder with open top is

 

πr² + 2πrh = 8

 

 

The volume of the shape is

 

V = πr²h

 

Substitute the value of h into the volume formula.  From the area formula,

 

h = (8 - πr²) / 2πr

 

 

V = πr²(8 - πr²) / 2πr

V = r(8 - πr²) / 2

V = 4r - (1/2)πr³

 

Now set the derivative of V equal to zero.

 

4 - (3/2)πr² = 0

 

Solve for r from this equation.  Then use test points to evaluate the derivative.  The derivative should change from positive to negative to get a minimum.  That will be the radius to give the maximum volume.

Answer 2

Because of the open top, the formula for the surface area is pi*r^2 + 2*pi*r*h = 8

 

 Solving for height, H = (8-pi*r^2)/(2*pi*r)  <---  call this equation H

 

Substituting this for the height in the formula for the volume of a cylinder:

 

  V = pi*r^2*H = pi*r^2 * (8- pi*r^2)/(2*pi*r) =  r/2*(8 - pi*r^2) = 4*r - pi/2*r^3

 

 Taking the derivative,  dV/dR = 4 - 3/2*pi*r^2

 

 Optimizing, 0 = dV/dR = 4 - 3/2*pi*r^2

 

  4 = 3/2*pi*r^2

 

 r^2 = 8/(3*pi)

 

 So the optimal radius is the square root of 8/(3*pi)   or just over 0.92131773

  The exact value is obtained by rationalizing the denominator by multiplying the top and bottom by 3*pi.

   Optimal R = 2/3*((square-root(6*pi))/pi

 

Plugging this result into the equation for height labeled above, ironically givens the same result

for the optimal height.

 

Therefore, the max volume is V= pi * (optimal R)^2 * (optimal H) =

 

                                                  pi * [( 2*square-root(6*pi)/(3*pi)]^3 =

 

                                                  

                                                 pi* (8 * (6*pi) * square-root(6*pi))/(27*pi^3)

 

                                                =   16*square-root(6*pi)/(9*pi) or approximately 2.45684728513