What is the method for solving a steel pipe around a hallway?

[whoareyou]

Homework Statement

Homework Equations

Trigonometric identities and differentiation

The Attempt at a Solution

It's pretty simple to solve this question when the hallway is a right angle.

Differentiating -6cscθ + 9secθ and setting it equal to zero and solving will yield the proper answer. But I don't know if the same method can be applied for this question? How would I start to solve this question?

 

[Office_Shredder]

The same method can, but not using the same function obviously. Do you understand where the equation -6cscθ + 9secθ comes from geometrically? The same geometric argument can construct a function for this other hallway that you need to minimize

 

[whoareyou]

For the right-angle turn, you look at the part of the pipe in each hallway, and add them together. So cosθ = 9/L1 and -sinθ = 6/L2. You can make similar triangles in the hallway of question, but I don't think θ is can be used as the angle for both triangles. And how does the pi/4 play a part in this?

 

[Office_Shredder]

For the right-angle turn, you look at the part of the pipe in each hallway, and add them together. So cosθ = 9/L1 and -sinθ = 6/L2. You can make similar triangles in the hallway of question, but I don't think θ is can be used as the angle for both triangles. And how does the pi/4 play a part in this?

Let θ be the angle that the pipe makes with the 12 foot hallway's wall. You'll have to do a little geometry (not much, just tracking some angles) to find out what angle the pipe makes with the 8 foot hallway's wall, then set up a similar equation to calculate how long the pipe is. The fact that we have an extra pi/4 angle is just going to change what the relationship between the angle with the 12 foot wall and the angle with the 8 foot wall are related

 

[whoareyou]

So this is what I came up with,

but the resulting theta values do not equate to the answer, which is 51.7762 ft.

 

[Office_Shredder]

Adding up all the angles on one half of the red line on the right should give me a total of $\pi$. But on the right half of the line you have
$$\pi/4 + \pi/4+\pi/4+\pi/4+\theta+\pi/2-\theta = 3\pi/2$$

So you have to double check those

 

[whoareyou]

Got it! It was θ - π/4 for the second triangle. Thanks :).