. There are two functions that have some interesting uses in the real world, and they are the hyperbolic trig functions a: :c :c _ a: cosh(gg;) = $1 sinh(:1:) = % The graph of the hyperbolic cosine is called a catenary, and models the shape of an object hanging by its own weight between two points. This can be found in nature in spiderwebs, vines, the rope at the movie theater, power lines, and the Gateway Arch in St. Louis (although it is inverted). Some suspension bridge cables have this shape, but the Golden Gate Bridge actually does not because the roadway is so heavy that it pulls the cables into a parabolic shape instead of a catenary. Let's explore the properties of these two functions. (a) Compute :7 (sin(:::)). (no work to show) (b) Compute c:1:l:(sinh(ac)). (c) Compute ddx (cos(;v)). (no work to show) (d) Compute (llm (cosh(:c)). (e) Determine what well known function is obtained from sinh(:r) + cosh(z). (f) Use algebra to show that the hyperbolic sine function sinh(m) has an analog double angle identity to its trig counterpart sin(..~:). Hint: Show that sinh(2:1:) = 2 sinh(m) cosh(:1:). (g) Show (using calculus) that y = cosh(aa:) is minimized at the middle of the interval [a, a], where a is a positive real number. . You are installing a pool in your backyard. You'd like to have a rectangular pool surrounded by a 4 foot cement walkway on each side of the pool. If you want a 1000 square foot pool, what dimensions of the pool will minimize the amount of area that the pool and walkway combined will take up in your backyard