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c) When a flexible cable of uniform density is suspended between two fixed points and hangs of its own weight, the shape y = f(x) of the cable must satisfy a differential equation d 2y dx2 = k s 1 + dy dx2 where k is a positive constant. Consider the cable shown in the Figure 1 below. Figure 1: Cable hanging between two points. i) Let z = dy dx in the differential equation. Solve the resulting first-order differential equa- tion (in z), and then integrate to find y. [6] ii) Determine the length of the cable.
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