Strength Of A Beam

**Joel** · June 11th 2009, 12:46 AM

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

**Amer** · June 11th 2009, 04:20 AM

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

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

24 is the radius of the base of the cylinder

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

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

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

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

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

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

I wish that help

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