OME 1966 Problem 4

You want to hang a weight $P$ so that it is $7$ m below a ceiling. To do this, it is suspended by means of a vertical cable attached to the midpoint $M$ of a chain hung by its ends from two points on the ceiling $A$ and $B$ distant from each other $4$ m. The price of the cable $PM$ is $p$ pta/m and that of the chain $AMB$ is $q$ pta/m. It is requested: a) Determine the lengths of the cable and the chain to obtain the lowest price cost of installation. b) Discuss the solution for the different values of the relation $p/q$ of both prices. (It is assumed that the weight is large enough to be considered rectile lines the chain segments $AM$ and $MB$ ).