# Thread: Need help in optimization

1. ## Need help in optimization

1. A ladder simultaneously leans against a 4.0m wall and a tall building that is 1.0 m behind the wall. Find the length of the shortest ladder that can be used. How high against the building will the leader reach in this case?

Do the optimization in terms of the angle that the ladder makes with the ground.

Diagram:

So basically I am asked to express both L1 + L2 in terms of theta where L = L1 + L2 and theta = u

Using similar triangles:
4/x = h/x+1

and:

L1 = 4/sin u
L = h/sin u
L2 = h/sin u - L1

If I am going about this correctly what should I do next? I mean I'm at a complete loss here as there are just too many unknowns and I know that in optimization you should reduce the variable to just one which would be theta. How would I go about in doing this?

2. using pythagoras' theorum

$\displaystyle L_1 = \sqrt{x^2 - 4^2}$

Using similar Triangles
$\displaystyle L = L_1 * \frac{1+x}{x}$

combine equations
$\displaystyle L = \frac{(1+x)\sqrt{x^2 - 4^2}}{x}$

Can you differenciate that (may want to simplify first)?

3. I need a function in terms of theta though and im not sure if the one i gave is correct. Also L1 = sqrt of x^2 + 4^2.

4. You have an expression in x (if u fix the pythag as you suggested)

You can rewrite this as a function in theta if you note that
$\displaystyle X= \frac{4}{tan(\theta)}$

i wouldn't like to differenciate that though!

5. here is an easier way, i modified your diagram a bit:

From this, you can see that the length of the ladder is:

$\displaystyle L = \frac{4}{sin(\theta)} + \frac{1}{cos(\theta)}$

You can differenciate that with respect to theta.