Optimization problems: Finding the optimal path

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EDIT

To solve your question:

The path you should take is a combination of both traveling on the sidewalk and through the grass.

Let's start:

$$T(x) = \frac{\text{Distance through grass}}{\text{Rate through grass}} + \frac{\text{Distance on sidewalk}}{\text{Rate on sidewalk}}$$

$$T(x) = \frac{2000-x}{6} + \frac{\text{Distance through grass}}{4}$$

And

$$600^2 + x^2 = (\text{Distance through grass})^2$$

$$\text{Distance through grass} = \sqrt{360000 + x^2}$$

$$T(x) = \frac{2000-x}{6} + \frac{\sqrt{360000 + x^2}}{4}$$

$$T'(x) = \frac{-1}{6} + \frac{x}{4\sqrt{360000 + x^2}} = 0$$

Solving, we get $x = 240\sqrt{5}$

And our distances are:

$$\text{Through Grass} = 360\sqrt{5}$$ $$\text{On sidewalk} = 2000 - 240\sqrt{5}$$

Comment if you need further explanation.

Also take a look at this, it might help.

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Irresponsible Newb
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Updated on August 01, 2022

Comments

  • Irresponsible Newb
    Irresponsible Newb over 1 year

    I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to combine two variables into a single variable and set up a related rates problem, but I'm not sure how to get started.

    The problem is as follows: Alaina wants to get to the bus stop as quickly as possible. The bus stop is across a grassy park, 2000 feet west and 600 feet north of her starting position. She can walk west along the edge of the park on the sidewalk at a speed of 6 ft/s. She can also travel through the grass in the park, but only at a rate of 4 ft/s. What path will get her to the bus stop the fastest?

    Could someone help set it up for me? Thanks!

    • dlev
      dlev over 9 years
      Hint: Suppose you walk some distance along the sidewalk before heading through the grass. Call that distance x. Now draw a diagram: can you get an expression for the distance through the grass in terms of x?
    • Varun Iyer
      Varun Iyer over 9 years
      Please look at my answer, I think you'll understand how to solve it now.
  • dlev
    dlev over 9 years
    Traveling along the hypotenuse is only faster if the speed along all sides is the same. It's not, though: traveling along the edge of the park is 1.5 times faster, so a combination of sidewalk and grass walking is optimal.
  • Varun Iyer
    Varun Iyer over 9 years
    @dlev I see that now, but OP wants to know whether traveling along the legs OR through hypotenuse is optimal. Therefore, even though the optimal distance is a combination of both, I cannot say this
  • dlev
    dlev over 9 years
    I don't think that's what the problem is asking. These problems appear in intro calc courses all the time, and they almost always want a combination of paths.
  • Varun Iyer
    Varun Iyer over 9 years
    @dlev Well I can change my answer then
  • Irresponsible Newb
    Irresponsible Newb over 9 years
    That link at the end was fantastic, much better than my math book, thanks!