In 1696, the first calculus textbook was published by theMarquis de lHopital. The

Chapter 4, Problem 108

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In 1696, the first calculus textbook was published by theMarquis de lHopital. The following problem is a simplifiedversion of a problem from this text.In Figure 4.122, two ropes are attached to the ceilingat points 3 meters apart. The rope on the left is 1meter long and has a pulley attached at its end. The ropeon the right is 3 meters long; it passes through the pulleyand has a weight tied to its end. When the weight isreleased, the ropes and pulley arrange themselves so thatthe distance from the weight to the ceiling is maximized.(a) Show that the maximum distance occurs when theweight is exactly halfway between the the pointswhere the ropes are attached to the ceiling. [Hint:Write the vertical distance from the weight to theceiling in terms of its horizontal distance to the pointat which the left rope is tied to the ceiling. A computeralgebra system will be useful.](b) Does the weight always end up halfway between theceiling anchor points no matter how long the lefthandrope is? Explain.