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A runner starting at a point P on a straight road runs east towards point Q, which is on the road and 8 kilometers from point P.  Five kilometers due north of point Q is a cabin.  She runs down the road for a while, at a pace of 12 kilometers per hour.  At some point Z between P and Q, the runner leaves the road and makes a straight line towards the cabin through the snow, hiking at a pace of 4 kilometers per hour.

 

Problem:  Determine where the runner should turn into the snowy woods in order to minimize the time to get from point P to the cabin.

 

If anyone know how to solve this problem and can give step by step instructions, I would really appreciate it.  Thanks.

 Feb 4, 2019
 #1
avatar+98129 
+2

Let P =  (0, 0)

Let Q = (8,0)

Let the cabin be at (8, 5)

 

And let ( x, 0)  be the point where the runner leaves the road  =  Z

 

The distance that the runner runs from P to Z = x

And the distance the runner ravels from Z to the cabin will be the hypotenuse of a right triangle with legs of 5  and (8-x)

 

And   Distance / Rate = Time

 

So....the total time, T, that the runner runs is

 

T =  x / 12  +  (5^2 + (8-x)^2 )^(1/2) / 4   simplify

 

T =  x /12  +  [ 25 + x^2 - 16x + 64 ] ^(1/2 ) / 4

 

T = x /12  +  [  x^2 - 16x + 89 ]^(1/2)  /  4         take the derivative  and set to 0

 

T '  = 1/12   + [ 2x - 16] / [8 √ [  x^2 - 16x + 89 ]^(1/2) ]   = 0

 

1/12  =   [ 16 -2x ] / [ 8 [  x^2 - 16x + 89 ]^(1/2)  ]

 

8 [ x^2 - 16x + 89 ]^(1/2) / 12   = 16 - 2x

 

(2/3)  [ x^2 - 16x + 89 ]^(1/2)   = 2 [ 8 - x]         

 

[ x^2 - 16x + 89]^(1/2)  = 3[8 - x ]       square both sides

 

x^2 - 16x + 89  = 9 [ x^2 - 16x + 64 ]

 

x^2 - 16x + 89  = 9x^2 - 144x + 576

 

8x^2 - 128x + 487 = 0

 

Solving this gives us 1 solution that is in the range of [ 0, 8]

This  is x ≈ 6.232  km   

 

So...the runner should run 6.232 km from P to Z    and then turn into the woods

And this will minimize the time from P to the cabin

 

 

 

cool cool cool

 Feb 4, 2019

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