A cable is suspended between two poles as shown in Fig-ure 6.9.2. Assume that the equation of the curve formed bythe cable isy=acosh(x/e), whereais a positive constant.Suppose that thex-coordinates of the points of support arex=-bandx=b, whereb>0.(e) Show that the lengthLof the cable is given byL=2asinhbe(b) Show that the says( the vertical distance between thehighest and lowest points on the cable) is given byS=acoshbe-a71-72These exercises refer to the hanging cable described in Exercise 70.071.Assuming that the poles are 400 ft apart and the sag in thecable is 30 ft, approximate the length of the cable by approx-imatinga. Express your final answer to the nearest tenth ofe foot. [ Hint:First letu=200/a.]72.Assuming that the cable is 120 ft long and the poles are 100ft apart, approximate the sag in the cable by approximative. Express your final answer to the nearest tenth of a foot. [Hint:First letu=50/a.] Suppose that a regionRin the plane is decomposedinto two regionsRiandRewhose areas areAnandA:, respectively, and whose centroids are( x1, ya)and( xz, ya),respectively. Investigate the problem of expressing the cen-troid ofRin terms ofA:, Az, ( x 1, y 1), and ( x 2, yz). Write ashort report on your investigations, supporting your reason-ing with plausible arguments. Can you extend your resultsto decompositions ofRinto more than two regions? Show: If the submarine in Exercise 20 descendsvertically at a constant rate, then the fluid force onthe window increases at a constant rate.(b) At what rate is the force on the window increasing ifthe submarine is descending vertically at 20 ft/min?22.(8) LetD=Didenote a disk of radiusasubmerged ine fluid of weight densitypsuch that the center ofDishunits below the surface of the fluid. For eachvalue ofrin the interval(0,a], I etDrdenote thedisk of radiusrthat is concentric withD. Select aside of the diskDand defineP(r) to be the fluidpressure on the chosen side ofDr. Use (5) to provethatlimr-0.P(r)=ph(b) Explain why the result in part (a) may be interpretedto mean thatfluid pressure at a given depth is thesame in all directions. (This statement is one ver-sion of a result known asPascal's Principle.) Approximate In 5 using the midpoint rule withn=10, andestimate the magnitude of the error by comparing your an-swer to that produced directly by a calculating utility .6.Approximate In 3 using the midpoint rule withn=20, andestimate the magnitude of the error by comparing your an-swer to that produced directly by a calculating utility