Calculating the quickest path between two points.

Question

Determine the pattern according to the situation, the quickest route from checpoint 1 checkpoint to 2 when the orienteer's running speed on the stomping is 2.5 times as large as on the swamp. X=750m and Y=400m

Answer 1

Hint: Pythagorean Theorem

Let $h$ denote the distance of the hypotenuse (from checkpoint 1 to checkpoint 2, through the swamp), where the legs are measured as horizontal distance (x) and vertical distance (y).

$$h^2 = (400)^2 + (750)^2 \implies h = \sqrt{ 400^2 + 750^2} =850\text{m}\,\text{through the swamp}$$

The alternative route to consider would be $400$ meters through the swamp (vertically), plus $750$ meters running on the stomping (horizontally).

Now, compare the two routes given the differences in running speed on stomping vs. on swamp.

Answer 2

Hint:

Lets assume you run in a straight line to a point $x$ from point 2 along the stomping path. Then run along the stomping path. How far do you travel and how long does it take.

Now from Pythagoras

Distance travelled in swamp is:

$$ \sqrt{ (750 - x)^2 + 400^2} $$

Distance travelled alond stomping path is:

$$ x $$

Now we don't know absolute speed but we know relative speed so time for the journey is proportional to.

$$ x + 2.5 \cdot \sqrt{ (750 - x)^2 + 400^2}$$

Your task is to find $x$ to minimise this time. What do you know about the derivative of a function at a maximum or minimum?