+1904 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.
+128408 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
