A rectangular sheet of perimeter of 36 cm and dimensions x cm by y cm is to rolled into a cylinder. what values of x and y give the largest volume

Given that perimeter of the rectangular sheet = 36cm, Height of cylinder = y base circumference of the cylinder = x = 2pir

So x+y = 36/2 = 18, y = 18-x, so height of the cylinder = 18-x and radius of cylinder = x/2pi

So volume of cylinder = pir^2h = pi (x/2pi) ^2 * (18-x) = (x^2 * (18-x)) / (4pi)

Let f (x) = (x^2 (18-x)) / (4pi)

We need to maximise this volume function.

So f' (x) = (1/4pi) (-x^2 + 2x (18-x)) = (1/4pi) (x) (-x+36-2x) = (1/4pi) (x) (-3x+36)

So f' (x) = 0⇒ (1/4pi) (x) (-3x+36) = 0⇒x=0 or x = 12

Since x+y = 18, so 0
f' (x) = (1/4pi) (-3x^2+36x)

So f'' (x) = (1/4pi) (-6x+36) ⇒ f'' (12) = (1/4pi) (-6*12+36) = - 9pi <0

So using second derivative test x=12 gives the maximum volume.

Since x+y = 18 so 12+y=18 ⇒y = 6

So x=12cm and y = 6cm give the largest volume.