# A rectangular beam is cut from a cylindrical log of radius 25 cm. The strength of a beam of width...

## Question:

A rectangular beam is cut from a cylindrical log of radius 25 cm. The strength of a beam of width w and height h is proportional to wh{eq}^{2} {/eq}. Find the width and height of the beam of maximum strength.

## Finding maxima and minima:

Given that we have a function f dependent on one variable x,

We can find the extremum of f by setting its derivative w.r.t x to zero i.e., {eq}\frac{df}{dx} = 0 {/eq}

Given that rectangular beam is cut from circular cross-section of radius r,

The diagonal of the rectangular beam passes through the center of the circle.

Also, {eq}w^2 + h^2 = {(2r)}^2\\ h^2 = 4r^2 - w^2 {/eq}

Let strength be given by S

{eq}S = kwh^2 {/eq} where k is proportionality constant.

{eq}S = kw(4r^2 - w^2) \\ S = k(4wr^2 - w^3) {/eq}

For maximum S,

{eq}\frac{dS}{dw} = 0\\ 4r^2 - 3w^2 = 0 \\ w=\frac{2r}{\sqrt{3}} \\ w = 28.8675 cm \\ h = 40.8248 cm {/eq}