A rectangular piece of sheet metal with a perimeter of $50$ cm is rolled into a cylinder with two open ends.

1. Write equations for the radius and height of the cylinder in terms of $x$.

   Hint (a):

   Look at the diagram of the cylinder. The circumference of the base is equivalent to $x$.

   Hint (a):

   $P = 2b + 2h$
   $25 = 2x+2h$
   Solve for $h$.

2. Express the volume of the cylinder as a function of $x$.

   Hint (b):

   $V=\pi\left(r^2\right)\left(h\right)$. Substitute.

3. Determine the value of $x$ that will maximize the volume and calculate the maximum value.

   Hint (c):

   The volume will be at a maximum when its derivative is $0$.