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!
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