[soze=3]Hello, dklee41![/size]

The stiffness of a rectangular beam is jointly proportional

to the breadth and the cube of the depth.

Find the dimensions of the stiffest beam that can be cut from a log

in the shape of a right-circular cylinder of radius $\displaystyle a$ centimeters. Code:

* * *
* *
*-------+-------*
*| : |*
| y: |
* | : | *
* | * | *
* | : \ | *
| y: \a |
*| : \ |*
*-------+-------*
* x x *
* * *

The breadth is $\displaystyle 2x$, the depth is $\displaystyle 2y.$

Since $\displaystyle S \:=\:kbd^3$, we have: .$\displaystyle S \:= \:k(2x)(2y)^3\:=\:16kxy^3$ **[1]**

From the diagram we have: .$\displaystyle x^2 + y^2 \:=\:a^2\quad\Rightarrow\quad y \:=\:\sqrt{a^2 - x^2}$ **[2]**

Substitute [2] into [1]: .$\displaystyle S \;= \;16kx\left[\left(a^2-x^2\right)^{\frac{1}{2}}\right]^3 \;= \;16kx\left(a^2-x^2\right)^{\frac{3}{2}}$

And **that** is the function we must maximize . . .