Calculus Question

Question

You are in a sea kayak 400m offshore from point A. You are along a straight sandy beach. A storm is brewing and you wish to go to a shelter 1000m down the beach from A. If you can paddle your kayak at 2 km/h and walk at 3km/h, how far down the shore should you beach your kayak to reach the shelter in the shortest amount of time? Give your answer to the nearest metre.

Answer 1

Let x = the distance down shore that you paddle.  The total distance you paddle is (x²+400²)^1/2.  Once on shore, the distance you walk is 1000 - x.  distance = rate*time so time = distance/rate:

 

time = (x²+400²)^1/2/(2 k/h) + (1000-x)/(3 k/h)

 

Take the derivative of time wrt x, set it to zero, then solve for x.