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5-14.
  1. A rectangular piece of sheet metal with a perimeter of 50 cm is rolled into a cylinder with two open ends. Homework Help ✎

    1. Find the radius and height of the cylinder in terms of x.

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

    3. Find the value of x that will maximize the volume and find the maximum value.

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

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

V = π(r²)(h). Substitute.

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