Find the Shortest Time for Woman to Reach Fitness Club

[thomasrules]

Homework Statement

A woman lives on an island 2km from the mainland. Her fitness club is 4km along the shore from the point closest to the island. To get to the fitness club, she paddles her kayak at 2km/h. Once she reaches the shore she jogs at 4km/h. Determine where she should land to reach her fitness club in the shortest possible time.

Homework Equations

The Attempt at a Solution

I set up a co-ordinate system with (0,0) the island @ (2,0) and the fitness club @ (0,4)

Not sure how to approach this one. How do I express an equation as a function of time?

 

[matt grime]

That would depend on the function.

But all you need to do is to work out the time taken if she lands a displacement X away from the nearest point, and the minimize that. it is easier if we draw a picture. I hope it works out.

-----C--------L-------N------------------------------I--------

If she lands at L what is her time to get the the club? This is a function of how far L is from N.

If she lands at N, then she has paddled two k, then has to run 4 k. If she kayaks to the club directly how long does it take?

 

[drpizza]

I'm not sure I'd bother with a coordinate system, but rather, with a picture. There are two simple solutions: kayak directly to the club (figure out the time it would take to do that) or kayak straight to shore (perpendicular) and then jog to the club. (calculate the time to do that too.) Then, there's a slightly tougher solution: Kayak to some point in between and jog the rest. You have two distances: the kayaking distance, and the running distance.
Two hints. If the total distance along the shore is 4 miles, and you land somewhere in between, that point is x miles from one end of that 4 miles, and 4-x miles from the other end. Hint 2: some greek guy, along time ago, used some formula for right triangles that the babylonians had already figured out; and he managed to get his name attached to that formula.

 

[HallsofIvy]

Let x be the distance from the "perpendicular point" to the place where she beaches her kayak. The distance she walks is 4- x km. The distance she paddles you can get from the Pythagorean theorem. Since you know her speeds walking and paddling, you can write her time as a function of x and the miminize that.

 

[3trQN]

The minimum distance is the distance directly to the fitness club, Kayak all the way. The maximum distance is direct to to shore, then to the club.

The optimum strategy is to maximise the time on land and minimise the distance travelled. The two are inversely proportional, find the saddle point.

There are a couple of nice things about his problem, i like it

Think about the optimum angle for launching ballistics, and why its optimum.