Originally Posted by

**galactus** #3.

We are told the strength is proportional to the product of the width and the square of the height(depth).

So, we can write $\displaystyle S=kwh^{2}$

S is the strength and k is a constant that depends on the wood type, etc.

We find a function that relates w and h. Draw a line from the center of the log to the corner of the beam.

We get a right triangle. $\displaystyle (\frac{w}{2})^{2}+(\frac{h}{2})^{2}=36$

$\displaystyle w^{2}+h^{2}=144$

$\displaystyle h^{2}=144-w^{2}$

Now, sub into the strength equation:

$\displaystyle S=kw(144-w^{2})=k(144w-w^{3})$

Now, differentiate, set to 0 and solve for w. h will follow.