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An ant travels from point A(0, -63) to point B(0, 74) as follows. It first crawls straight to (x, 0) with \(x \ge 0\), moving at a constant speed of  sqrt2 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?

 

Thank you very much!

:P

 Feb 25, 2019
 #1
avatar+98091 
+2

Here's my best attempt

 

Time =   [distance from (0, -63) to (x , 0)] / rate    plus [ distance from (x, x) to (0, 74) ] / rate 

 

Time  =  √ (x^2 + 63^2) / √ 2  +  √ [x^2 + (74 - x)^2 ] / 2      

 

We could use Calculus to solve this, but a graph seems easier

 

Here is a graph of the function : https://www.desmos.com/calculator/mmudrsppbh

 

It shows that the time is minimized when x  ≈ 23.3

 

And the minimized time ≈ 75.4 sec

 

 

cool cool cool

 Feb 25, 2019
edited by CPhill  Feb 25, 2019
 #2
avatar+580 
+2

Yes, you are right! Thank you!

 Feb 25, 2019
 #3
avatar+98091 
0

I like this problem.....!!!

 

Were you assigned this or did you make it up   ????

 

 

cool cool cool

CPhill  Feb 25, 2019

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