Question
A rectangular beam is to be cut from a log with a circular cross section. If the strength of the beam is proportional to its width and the square of its depth, find the dimensions which give the strongest beam. Let a be the (constant) diameter of the log. Then the strength S of the beam is S=kwd^2, where i is constant. Note the diameter of the log is equal to the diagnol measurement of the beam's cross section, so w^2 + d^2 = a^2. Find the dimensions of the beam (d and w) that maximize its strength S.
I honestly don't even know where to start with this one.
Explanation / Answer
Let the width = b and depth = h (standard notation of beam section in Strength of Materials in Mechanical Engineering). What you refer to as 'strength' is actually a factor of 2nd moment of area of the cross section. Let us call it strength, for your sake. So, strength, say S, = k*b*d^2 where k = constant of proportionality (1/12 for rectangular beam). Now for the beam to be of maximum strength the dimensions of cross section should be as large as possible. Hence, the rectangle has vertices lying on the circle with 20 in diameter. Hence, d^2 + b^2 = 20^2 [by Pythagorus' theorem]. So, S = kb(400 - b^2) dS/db = k(400 - 3b^2) d^2S/db^2 = -6kb < 0 So S is max when dS/db = 0 i.e. when 400 - 3b^2 = 0 b = 20/v3 So, d = 20v(2/3). Hence the strongest rectangular beam cross section is width = 20/v3 = 11.55 inches depth = 20v(2/3) = 16.33 inches