Maximizing Pipe Length: Solving a Horizontal Carry Dilemma

[Slimjimjohnso]

A steel pipe is being carried horizontally down a hallway that is 9ft wide. At the end of the hallway there is a right turn into a hallway that is 6ft wide. What is the length of the longest pipe that can be carried horizontally around the corner? (Hint: minimize the line)

I have looked at this problem and tried many ways to solve it. I cannot think of a way to relate my information to make a derivable function.

I did make a bunch of triangles and using geometry I'm pretty sure the answer is 21ft but man if I can't do this with calculus. Help!

 

[xcvxcvvc]

A steel pipe is being carried horizontally down a hallway that is 9ft wide. At the end of the hallway there is a right turn into a hallway that is 6ft wide. What is the length of the longest pipe that can be carried horizontally around the corner? (Hint: minimize the line)

I have looked at this problem and tried many ways to solve it. I cannot think of a way to relate my information to make a derivable function.

I did make a bunch of triangles and using geometry I'm pretty sure the answer is 21ft but man if I can't do this with calculus. Help!

21ft? That's longer than both hallway's are wide put together.

 

[Slimjimjohnso]

Imagine an L shaped hallway. The short part of the L is 9ft the long part is 6ft. If you drag one side along the 9ft part around the corner into the 6ft part why couldn't it be 21ft?

I would show the picture I have drawn but I am on my phone at work and don't have a way to post it here.

 

[Slimjimjohnso]

http://lh4.ggpht.com/_KCqQzK7M3K4/TEJcWqbbWxI/AAAAAAAAAAU/lyWzaS6BdFI/s400/2010-07-17 20.27.40.jpg

There that should be the picture with my triangles.

I broke the left half into 3 right trianlgs joined by a equilatteral. I solved for the hypotenuse of one and since its part of a bigger triangle all sides are even. I then solved the big triangle consisting of both hallways for its hypotenuse getting 21.

I need help doing this the calc way. Any direction would be awesome.

 

[xcvxcvvc]

http://lh4.ggpht.com/_KCqQzK7M3K4/TEJcWqbbWxI/AAAAAAAAAAU/lyWzaS6BdFI/s400/2010-07-17 20.27.40.jpg

There that should be the picture with my triangles.

I broke the left half into 3 right trianlgs joined by a equilatteral. I solved for the hypotenuse of one and since its part of a bigger triangle all sides are even. I then solved the big triangle consisting of both hallways for its hypotenuse getting 21.

I need help doing this the calc way. Any direction would be awesome.

The problem states that the steel pipe begins by being carried horizontally down a hallway that is 9 ft wide... sorry but this is just a horrible wording of the question if that diagram was the setup in mind.

 

[Slimjimjohnso]

The problem and picture is directly off of my homework.

I added the lines in between the L shaped hallway. But the line is the hypotenuse of a bigger triangle. The only thing I can think to relate the problem to calculus is through some sort of trig identities.

 

[vela]

Break the length of the pipe into two pieces, x₁ and x₂, where x₁ is the portion that's in the 9' hallway and x₂ is the portion in the 6' hallway. Relate these lengths to the angle θ and the width of the respective hallway. You'll get an expression for the total length of the pipe L=x₁+x₂ as a function of θ. That's the function you want to minimize.

 

[Slimjimjohnso]

Thank you very much for your help.

I ended up with

I took cos(theta)=9/a and sin(theta)=6/b
To get my function

L(theta)=9sec(theta)+6csc(theta)