The stiffness S of a rectangular beam is proportional to its width times the cube of its depth.

a. Find the dimensions of the stiffest beam that can be cut from a 12-in.-diameter cylindrical log.

b. Graph S as a function of the beam’s width w, assuming the proportionality constant to be k = 1. Reconcile what you see with your answer in part (a).

c. On the same screen, graph S as a function of the beam’s depth d, again taking k = 1. Compare the graphs with one another and with your answer in part (a). What would be the effect of changing to some other value of k? Try it.