Optimization/ Newtons Method

[leijonah]

A steel pipe is being carried down a hallway 9 ft. wide. At the end of the hall there is a right angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?
Can someone help me with where to start with this problem??

Also this is tricky

Which is bigger e^pie or pie^e without using calculators, by using maybe newtons method?
So is this right for equating it too x.
f(x)=(x)^(pie/2)-e=0 for e^pie
f(x)=(x)^e/2-pie=0 for pie^e

 

[leijonah]

Anybody any insight???

 

[galactus]

This is assuming the diameter of the pipe is not be considered?. That makes the problem a little more complicated. So, assuming diameter is not an issue.

We'll use similar triangles.

The maximum length of the pipe will be the smallest value of L=x+y

\(\displaystyle \frac{y}{9}=\frac{x}{sqrt(x^{2}-36)}=\frac{9x}{sqrt(x^{2}-36)}\)

\(\displaystyle L=x+\frac{9x}{sqrt(x^{2}-36)}\)

Now differentiate, set to 0 and solve for x.

Something I forgot. There is a formula, believe it or not, for this sort of problem. It can be derived from a similar method. Use the known hallway widths.

Let a=9 and b=6

\(\displaystyle (a^{\frac{2}{3}}+b^{\frac{2}{3}})^{\frac{3}{2}}\)

You should get the same answer as you got above with the derivative.