MHB Revisit t the ladder optimization problem

[karush]

This is a common homework problem but..

A fence $6$ ft high runs parallel to the wall of a house of a distance of $8$ ft
Find the length of the shortest ladder that extends from the ground,
over the fence, to the house of $20$ ft high
and the horizontal ground extends $25$ ft from the fence.

$$L=\sqrt{\left(x+8 \right)^2 +\left(20-y\right)^2 }$$

its assumed implicit derivative but really?

I graphed this and noticed the local min was about $19.7$

$$\sqrt{\left(x+8 \right)^2 +\left(\frac{48+6x}{x}\right)^2 }$$

 

[MarkFL]

I used Lagrange multipliers to derive a formula here:

https://mathhelpboards.com/questions-other-sites-52/his-question-yahoo-answers-regarding-finding-shortest-ladder-will-reach-over-fence-9939.html

$$L_{\min}=\left(d^{\frac{2}{3}}+h^{\frac{2}{3}} \right)^{\frac{3}{2}}$$

Plugging in the given data:

$$d=8\text{ ft},\,h=6\text{ ft}$$

We find:

$$L_{\min}\approx19.73\text{ ft}$$