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An ant travels from the point A(0,-63) to the point B(0,74) as follows. It first crawls straight to (x,0) with x>=0, moving at a constant speed of sqrt(2) units per second. It is then instantly teleported to the point (x,x). Finally, it heads directly to B at 2 units per second. What value of x should the ant choose to minimize the time it takes to travel from A to B?

 

I was able to get to \(Total time = \frac{\sqrt{x^2+\left(74-x\right)^2}}{2}+\frac{\sqrt{x^2+63^2}}{\sqrt{2}}\)

 

How do I find x such that it take "t" least amount of time

 Jan 15, 2021
edited by geoNewbie21  Jan 15, 2021
 #1
avatar+118588 
+1

You have the correct Total Time function

 

We could use Calculus  to solve this but the  derivative is a little messy....I solved it  here  with a graph :

 

https://web2.0calc.com/questions/interesting-geometry

 

However...if you want me to, I  think I can solve it with pure Calculus.....

 

cool cool cool

 Jan 15, 2021
 #2
avatar+122 
+1

Yes I did see your solution as 23.31 is the value of x. I assume that it would have some rounding because of sqrt. I am trying to get to the value of x in terms of sqrt root so I do not make any approximations. Would calculus help with that? 

 

With calculus would it be d/dt(above function) -> 0? Could you please help with calculus

 Jan 15, 2021
 #3
avatar+118588 
+1

T =   ( x^2  + (74 - x)^2 )^(1/2) / 2  +  ( x^2 + 63^2) / sqrt (2)

 

T  =   ( x^2  + x^2 - 148 + 5476 )^(1/2) / 2   +  ( x^2 + 3969)^(1/2) / sqrt (2)

 

T'     =   (1/4) ( 4x - 148)/ ( 2x^2 - 148x + 5476)^(1/2)  +  (2x) /[sqrt (2)(x^2 + 3969)^(1/2)]    = 0

 

       ( x - 37)                                   x

_____________________  +  ______________________     = 0  

(2x^2 - 148x + 5476)^(1/2)        sqrt (2) (x^2 + 3969)^(1/2)

 

Note  that  we  can write  the denomnator of the  first fraction as

 

2 ( x^2 - 74x + 2738)

 

So   we can write

 

(x - 37)                                                                    x

__________________________   +  _____________________   =   0

sqrt (2) ( x^2 - 74x + 2738)^(1/2)       sqrt (2) ( x^2 + 3969)^(1/2) 

 

 

 

(x - 37)                                                          -x

____________________    =              _________________           sqaure both sides

(x^2 - 74x + 2738)^(1/2)                        (x^2 + 3969)^(1/2) 

 

 

(x - 37)^2                                  x^2

_______________     =   ___________       cross-multiply

x^2 - 74x + 2738                x^2 + 3969 

 

 

(x - 37)^2 ( x^2 + 3969)  =   x^2 ( x^2 - 74x + 2738)     simplify

 

x^4 - 74 x^3 + 5338 x^2 - 293706 x + 5433561   =  x^4 - 74 x^3 + 2738 x^2

 

After a little manipulation we get

 

2600 x^2 - 293706 x + 5433561 = 0

 

Putting this into the quadratic formula  and  evaluating  we get  that  

 

 

x =   [ 293706   - sqrt  ( 293706^2  - 4*2600*5433561)]  / (5200)  =

 

[ 293706  -  sqrt ( 29754180036)  ]  / 5200  =

 

[ 293706  - 172494  ] /5200  =   23.31  / 100  =   23.31  = x

 

(There is another  value of x but it does not give us a minimum.......this is probably  due to the  fact that we squared the derivative )

 

 

cool cool cool

 Jan 15, 2021
 #4
avatar+122 
+1

Thank you. Got it.

geoNewbie21  Jan 15, 2021
 #5
avatar+118588 
0

OK...glad to help....

 

 

cool cool cool

CPhill  Jan 16, 2021

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