It's about Hyperbolic Functions.

Letter to a Friend: (5 points). As usual, explain your solutions to the following problems in the form of a letter to a friend. The ends of a chain of uniform density are attached to the tops of two vertical poles with equal heights that are placed 2b feet apart. The chain hangs in equilibrium in a uniform gravitational eld, with the only other external forces being the two exerted by the poles on the end of the chain. With a, b > U, and c E R constants having units of feet, the function h = f(.v) = acosh(:t/a) + c, b S a: g b, expresses the height above the ground (in feet) of the point P of a chain in terms of its horizontal displacement (in feet) from the chain's low point. (a) Start by explaining the role of the hyperbolic cosine function in de- scribing chains of uniform density that hang in a uniform gravitational eld. Part of your explanation should include an description of the roles played by the parameters a and c in the height function above. (b) Find a function h = g(o:) that expresses the height of a point P on the chain in terms of the acute angle (measured in radians) the chain makes with the horizontal at P. Simplify the function. State its domain. Your solution should not appeal to any inverse hyperbolic function and the output of your nal function should be a very simple expression in terms of the variable 0: and the parameters a and b