Question

A rectangular beam is cut from a cylindrical log of radius 25 cm. A cross-section of the log is shown below. The strength, S, of a beam of width w and height h is proportional to wh2.
(a) Express the strength of the beam as a function of w only. Use k as the proportionality constant.
S(w) =

(b) Find the exact width and height of the beam of maximum strength.
Width =
Height =

Answer 1

We are given the statement that the strength, S, of a beam of width w andheight h is proportional to wh2. hence using k as the proportionality constant, the equation is  S = k wh2. the exact width and height can be determined by applying optimization to the problem. optimization is taking the first derivative of an equation and equating then to zero