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5-14.

A rectangle with length x is rolled such that the circumference of the resulting tube is x. A rectangular piece of sheet metal with a perimeter of 50 cm is rolled into a cylinder with two open ends. Homework Help ✎

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

    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.

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

    V = π(r²)(h). Substitute.

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

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