Question: In 1696, the first calculus textbook was published by the Marquis de l'Hopital. The following problem is a simplified version of a problem from this text. In Figure, two ropes are attached to the ceiling at points meters apart. The rope on the left is 1 meter long and has a pulley attached at its end. The rope on the right is 3 meters long; it passes through the pulley and has a weight tied to its end. When the weight is released, the ropes and pulley arrange themselves so that the distance from the weight to the ceiling is maximized.
(a) Show that the maximum distance occurs when the weight is exactly halfway between the the points where the ropes are attached to the ceiling. [Hint: Write the vertical distance from the weight to the ceiling in terms of its horizontal distance to the point at which the left rope is tied to the ceiling. A computer algebra system will be useful.]
(b) Does the weight always end up halfway between the ceiling anchor points no matter how long the lefthand rope is? Explain.
