Optimizing Hiker's Route to Forest Cabin

  • Thread starter PAstudent
  • Start date
In summary: I don't think so.In summary, the hiker can walk a distance of x down the road before cutting through the forest, straight to the cabin. The trip will take the least time if x is equal to or greater than 3mi.
  • #1
PAstudent
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Homework Statement



A hiker starting at point P wants to get to a forest cabin, which is two miles from the road at point Q (See diagram). The hiker can walk at 6.00 mph on the road and 2.50 mph through the forest. Suppose the hiker walks a distance X down the road before cutting through the forest, straight to the cabin. Determine the value of X for which the trip take the least time. Consider the distances to be good to three significant figures.

upload_2015-8-27_16-2-24.png

Homework Equations

[/B]
Trigonometry and Derivatives

The Attempt at a Solution


I attempted to draw the lines again and label points to simplify the triangle. However, I am getting stuck on how to optimize the walk.
 
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  • #2
PAstudent said:

Homework Statement



A hiker starting at point P wants to get to a forest cabin, which is two miles from the road at point Q (See diagram). The hiker can walk at 6.00 mph on the road and 2.50 mph through the forest. Suppose the hiker walks a distance X down the road before cutting through the forest, straight to the cabin. Determine the value of X for which the trip take the least time. Consider the distances to be good to three significant figures.

Homework Equations

[/B]
Trigonometry and Derivatives

The Attempt at a Solution


I attempted to draw the lines again and label points to simplify the triangle. However, I am getting stuck on how to optimize the walk.

You really need to make an effort when posting schoolwork questions here.

You are given the speed for both paths. Write an equation for the total time taken based on those speeds and different values of x. Please show your work.
 
  • #3
I'm sorry I am awful at physics and my 70 year old professor doesn't teach anything. Just assigns chapters to read and gives us the basic formulas. I hope you have a great day
 
  • #4
PAstudent said:
I'm sorry I am awful at physics and my 70 year old professor doesn't teach anything. Just assigns chapters to read and gives us the basic formulas. I hope you have a great day

I'm sorry you are giving up so easily. It just takes some practice to get comfortable with these types of problems.

In case you can give it one more try, use my advice from my previous post about writing the equation for the time taken based on 2 paths and 2 speeds. Then plug in some numbers to see what the total time looks like for a few sample choices of x. Like if x=0, then it's just the diagonal path length at the woods-walking speed. And if x = 3mi, then the total time is formed by walking 3mi at the path speed and 2mi at the woods speed. The quickest time is probably somewhere in the middle of those two choices...
 
  • #5
PAstudent said:
I'm sorry I am awful at physics and my 70 year old professor doesn't teach anything. Just assigns chapters to read and gives us the basic formulas. I hope you have a great day
It can be really difficult when you have a teacher who doesn't give much apart from chapters in a book. The effect teachers can have on someones passion for a subject is huge
 
  • #6
That is the issue. I just freeze up when trying to write equations. I just don't know what to do. I know you guys hate hearing that and see it as a lack of effort.
 
  • #7
PAstudent said:
That is the issue. I just freeze up when trying to write equations. I just don't know what to do. I know you guys hate hearing that and see it as a lack of effort.
Add two more labels to the diagram. Label the point where the walker leaves the road A. The point on the road that is nearest to point Q label point R.
I want you to use symbols for the given values, not the numbers. Use PR instead of '3 miles' and QR instead of '2 miles'. Use u instead of 6 mph and v instead of 2.5mph. This is a useful habit to get into. You already have x for the (unknown) distance walked on the road.

Try to answer these questions in sequence:
In terms of these symbols, how long will the walker be on the road?
How far will the walker go through the forest? How long will that take?
 
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  • #8
So the how long walk on the road= (PR-x)/u?
 
  • #9
time = distance/velocity

So
time = x/u
 
Last edited:
  • #10
So to find how far walked through the forest would it be [sqrt(QR^2+AR^2)]/v which would find that straight diagonal line? Or would I have to use angles?
 
  • #11
PAstudent said:
So to find how far walked through the forest would it be [sqrt(QR^2+AR^2)]/v which would find that straight diagonal line? Or would I have to use angles?

Pythagoras works fine. No need for trig.

So now you have expressions for the time of each leg, the total time is just the sum of these.
Note that you'll need to express AR in terms of x.
You mention derivatives in the OP so I assume you know about minimisation problems?
 
  • #12
I understand derivatives, but I never really used them with minimization problems. How would you go about using a derivative to do that?
 
  • #13
PAstudent said:
I understand derivatives, but I never really used them with minimization problems. How would you go about using a derivative to do that?
If you have an independent variable x and a function of it f(x) that you want to find either a local minimum or local maximum of, you look at the derivative f'(x) of f (df/dx). At a point that is locally a minimum, a small change in x either way will increase f, so f' will be negative on one side of x and positive on the other. Right at the minimum, f' will be zero.
 

What is the purpose of optimizing a hiker's walk?

The purpose of optimizing a hiker's walk is to improve their overall hiking experience by finding the most efficient and comfortable way for them to walk. This can help prevent fatigue and injuries, and allow the hiker to reach their destination in a timely manner.

What factors should be considered when optimizing a hiker's walk?

Some of the factors that should be considered when optimizing a hiker's walk include the terrain, weather conditions, the hiker's physical abilities, and the weight of their backpack. It is also important to consider any pre-existing injuries or conditions that may affect their walking.

How can technology be used to optimize a hiker's walk?

Technology can be used to track and analyze a hiker's movement and provide real-time feedback on their walking technique. This can include wearable devices, such as fitness trackers, and mobile apps that use GPS to track the hiker's route and provide suggestions for improvement.

What are some techniques that can help optimize a hiker's walk?

Some techniques that can help optimize a hiker's walk include proper posture and body alignment, using trekking poles to distribute weight and provide stability, taking breaks and stretching regularly, and adjusting the backpack to evenly distribute weight on the hiker's body.

Are there any potential risks or drawbacks to optimizing a hiker's walk?

While optimizing a hiker's walk can have many benefits, there are potential risks and drawbacks that should be considered. For example, focusing too much on optimizing speed and efficiency may lead to neglecting proper rest and hydration, which can increase the risk of fatigue and injuries. It is important to find a balance and prioritize the hiker's safety and well-being above all else.

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