Expert: Paul Klarreich - 1/11/2008

Question
hello again. thank you for your help for the other question!

this question confuses me..

01. a man is in a boat 2 miles from the nearest point on the coast. he is to go to a point Q, 3 miles down the coast and i mile inland. if he can row at 2 miles per hour and walk at 4 miles per hour, toward what point on the coast should he row in order to reach point Q in the least time?

>>>all i could figure out was that i'm suppose to minimize the time so i would use this equation for the boat and the walking: t = d/r... and i'm not so sure what to do after figuring out that equation either

thank you once again!

Answer
Questioner:   alex
Category:  Calculus
Private:  No
 
Subject:  minimum time
Question:  hello again. thank you for your help for the other question!

this question confuses me..

01. a man is in a boat 2 miles from the nearest point on the coast. he is to go to a point Q, 3 miles down the coast and 1 mile inland. if he can row at 2 miles per hour and walk at 4 miles per hour, toward what point on the coast should he row in order to reach point Q in the least time?

>>>all I could figure out was that I'm supposed to minimize the time so I would use this equation for the boat and the walking: t = d/r... and i'm not so sure what to do after figuring out that equation either

thank you once again!
..................................
Hi, Alex,

You want to minimize the TOTAL TIME.  Split that into two parts:

Tw = walking time.
Tr = rowing time.

Now make a diagram.  He is at M, and wants to reach Q.  So he aims the boat at P:


|
+                       Q(3,1)  
|
|
|          P(x,0)  
-+-------+--*----+-------+----- Coast
|       1       2       3
|
|
|
M(0,-2)


MP = sqrt(x^2 + 4)

PQ = sqrt((x-3)^2 + 1)

Row MP in Tr = MP/2 = sqrt(x^2 + 4)/2

Walk PQ in Tw = PQ/4 = sqrt((x-3)^2 + 1)/4

Total time is the sum of those;

T = sqrt(x^2 + 4)/2 + sqrt((x-3)^2 + 1)/4

Now do the usual stuff:

              2x                2(x - 3)
dT/dx =  ---------------- + --------------------
        4 sqrt(x^2 + 4)    8 sqrt((x-3)^2 + 1)

              x                 (x - 3)
dT/dx =  ---------------- + --------------------
        2 sqrt(x^2 + 4)    4 sqrt((x-3)^2 + 1)


Set that equal to zero:


             x                 (x - 3)
0  =  ---------------- + --------------------
       2 sqrt(x^2 + 4)    4 sqrt((x-3)^2 + 1)


        2x sqrt((x-3)^2 + 1) +  (x - 3)sqrt(x^2 + 4)
0  =    -------------------------------------------
        4 sqrt(x^2 + 4)  sqrt((x-3)^2 + 1)

Set the top to zero:

2x sqrt((x-3)^2 + 1) +  (x - 3)sqrt(x^2 + 4) = 0

2x sqrt((x-3)^2 + 1) =  (3 - x)sqrt(x^2 + 4)

Square both sides:

4x^2((x - 3)^2 + 1) = (9 - 6x + x^2)(x^2 + 4)

4x^2(x^2 - 6x + 9 + 1) = (9 - 6x + x^2)(x^2 + 4)

4x^4 - 24x^3 + 40x^2 = 9x^2 - 6x^3 + x^4 + 36 - 24x + 4x^2

3x^4 - 18x^3 + 27x^2 =   + 36 - 24x

3x^4 - 18x^3 + 27x^2 + 24x - 36 = 0

x^4 - 6x^3 + 9x^2 + 8x - 12 = 0

x = 1 is an obvious solution:


1  - 6   9   8   - 12 (1)

    1  -5   4     12
------------------------
1  - 5   4  12     zero


x^3 - 5x^2 + 4x + 12 = 0 is the reduced equation.

A quick-and-dirty graph of x^3 - 5x^2 + 4x + 12  shows only a root for a negative number.

So x = 1 is our solution.