1. ## Applying derivatives

Suppose you are swimming in a lake and find yourself 200m from shore. You would like to get back to the spot where you left your towerl, which is at least 200m down the beach, as quickly as possible. You can walk along the beach at 100m/min, but you can swim at only 50m/min. To what point of the beach should you swim to minimize your total travelling time?

Is this function correct?

T=(200-x)/100 + (sqrt(200^2+x^2))/50

2. i am getting

t(x) = x/100 + (((200-x)^2 + 200^2)^1/2) /50)

as we can see if the guy swims all the way he will be going 80 000^1/2 at 50 m/min and will take him

5.66 hours appx

try x = 0 in both equations...

here i am using the starting point at the bottom left of the diagram and the x value is the horizontal distance from the towel to the point that the swimmer reaches shore

let me know what you think

3. one second you might be right... let me try x = 200 for yours

4. yeah okay i see what you did, we just had the diagram drawn differently

5. im getting that he should swim to the point approximately 84.55 metres away from the towel

i cant find the root of the derivative but i approximated a value very near to it

what did you get?

6. I tried to find the derivative, got this:

-1/100 + 1/(25(sqrt[200^2+x^2]))

I seem to get a negative value for x^2 which doesn't make sense.

7. differentiating yours i get -1/100 + ( x / (50(200^2 + x^2)^1/2)

try x = 84.55 in it and you will get -0.0022

and try 200-84.55 which would be 84.55 away from the towel and you get -0.000013

i think there might be two optimal positions for him to land on the shore