Question

The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a.

Answer 1

Let us assume ABCD be the cross section of beam that need to cut from the circular log of radius ‘a’. 

Let us assume ‘b’ be the depth of the rectangle, ‘l’ be the length of the rectangle and ‘d’ be the diameter of circular log. 

⇒ d = 2a …… (1) 

According to the problem, strength of the beam is given by ⇒ S = lb² …… (2) 

From the ΔABC, 

⇒ d² = l² + b² 

From (1) 

⇒ (2a)² = l² + b² 

⇒ b² = 4a² - l² …… (3) 

From(2) and (3) 

⇒ S = l(4a² - l²) 

⇒ S = 4a²l - l³ 

We need strength of the beam to be maximum, let us take S as a function of l 

For maxima and minima,

Differentiating S again,

We get maximum strength for length I = \(\frac{2a}{\sqrt3}\),

Lets find the depth ‘b’ for this l :

Let’s find the strength of beam for  I = \(\frac{2a}{\sqrt3}\) and b = \(\frac{2\sqrt2a}{\sqrt3}\)

The dimensions of beam of maximum strength is (\(\frac{2a}{\sqrt3},\frac{2\sqrt2a}{\sqrt3}\)).