# Thread: Strength Of A Beam

1. ## Strength Of A Beam

Hi All,

Not sure if this is in the right section, but would love your help with this.

Engineers have determined that the strength s of a rectangular beam varies as the product of the width w and the square of the depth d of the beam, that is, s = kwd² for some constant k. Find the dimensions of the strongest rectangular beam that can be cut from a cylindrical log with diameter 48cm.

Cheers

2. Originally Posted by Joel
Hi All,

Not sure if this is in the right section, but would love your help with this.

Engineers have determined that the strength s of a rectangular beam varies as the product of the width w and the square of the depth d of the beam, that is, s = kwd² for some constant k. Find the dimensions of the strongest rectangular beam that can be cut from a cylindrical log with diameter 48cm.

Cheers
find d with respect to w or find w with respect to d then sub it in the equation after that find the derivative and see the extreme max point I think the relation between them can given by

$\displaystyle 24^2=\frac{d^2}{2^2} + \frac{w^2}{2^2}$

24 is the radius of the base of the cylinder

$\displaystyle d^2=4( 24^2 - \frac{w^2}{4} )$

$\displaystyle d=2\left(\sqrt{24^2-\frac{w^2}{4}}\right)$ sub this in the equation of s

$\displaystyle s=kw\left(2\left(\sqrt{24^2-\frac{w^2}{4}}\right)\right)^2$

$\displaystyle s=4kw\left( 24^2 - \frac{w^2}{4} \right)$

find the derivative of s to find the extreme value that will be w then sub it in this equation to find d

$\displaystyle 24^2=\frac{d^2}{2^2} + \frac{w^2}{2^2}$

I wish that help

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