how do you solve ( 50 sin h( X/50 ) ] 33-33

Question

Suppose that you want to desing a  ski lift and need to determine how much cable will be needed between two posts that are 66 feet apart. the formula for the cable is  y= 50 cos h ( x/50 )

Answer 1

cosh means hyperbolic cosine, cosh(x/50)=½(e^x/50+e^-x/50).

y=50cosh(x/50) where x is the distance between the posts, and y is the height in feet above ground of the cable when suspended. So at its lowest point the cable is 50 feet above the ground

The cable forms a catenary (curve) and we need the (arc) length of the curve between x=-33 and x=33.

dy/dx=y'=sinh(x/50) where sinh(x/50)=½(e^x/50-e^-x/50).

For an infinitesimal length of arc ds, we have ds²=dx²+dy², so (ds/dx)²=1+(dy/dx)².

We can write this s'²=1+y'², so ds=√(1+y'²)dx and:

S=_-33∫³³√(1+y'²)dx=_-33∫³³√(1+sinh²(x/50))dx.

sinh(x/50)=½(e^x/50-e^-x/50), so sinh²(x/50)=¼(e^x/25-2+e^-x/25) and:

1+sinh²(x/50)=1+¼(e^x/25-2+e^-x/25)=¼(e^x/25+2+e^-x/25)=cosh²(x/50).

[From this we can see that the derivative of sinh(A))=cosh(A) and vice versa. Also cosh²(A)-sinh²(A)=1.]

Cable length S=_-33∫³³cosh(x/50)dx=50[sinh(x/50)]_-33³³. Note: sinh(-A)=-sinh(A).

S=100sinh(0.66)=70.897ft approx.