Working out a formula for 2 variables?

[shipscat]

Hi Folks,

I'm working through a pre-selection maths course and have come to a complete halt with the following Differentiation problem:-

" If the strength of a rectangular beam of wood varies as its breadth and the square of its depth, find the dimensions of the strongest beam that can be cut out of a round log, diameter d. "

It's the only problem in the notes that has more than 1 variable and no example to show how its done.

Here's what I've come up with so far...
S varies with Breadth, therefore S=kB
S varies with Depth^2, therefore S=kD^2
S=kBD^2
The derivative of a constant term = 0

If I could work out the formula I think I'd be ok, but I'm completely at a loss for what to do next.

Thanks for any help you can give me, and sorry it's a little basic.

Shipscat

 

[Hootenanny]

Hint: Can you related B and D to d?

 

[shipscat]

Hi Hootenanny,

I think I tried this last night, I came up with

B=d-2x
D=d-2y
therefore S=(d-2x)(d-2y)(d-2y)

Still with 2 variables. I'm assuming x and y cannot be equal as the problems states a rectangular beam.

Am I along the right lines or 90 degrees to them?

 

[Hootenanny]

Your on the right lines. Draw yourself a semi-circle, mark on your diameter together with B and ¹/₂D as chords. Now, join the two endpoints of B and ¹/₂D with a further chord. Now, what is the maximum size of this chord?

 

[shipscat]

I'm really, Really sorry Hootenanny, I was right with you up until " together with 1/2B and 1/2D as chords. " Where on the semi-circle should I be marking the chords? Do you mean 1/2 Breadth or 1/2 the distance from the edge of the beam to the circle?

I'm sorry to seem so dumb, but I really am dumb.