Question

You want to get from a point a on the straight shore of the beach to a buoy which is 60 meters out in the water from a point b on the shore. b is 70 meters from you down the shore. if you can swim at a speed of 3 meters per second and run at a speed of 5 meters per second, at what point along the shore, x meters from b, should you stop running and start swimming if you want to reach the buoy in the least time possible?

Answer 1

The time t for a constant speed is given by:
t = distance/ velocity

Let's call the distance to run on the beach x₁ and the remaining distance to swim x₂, then the equation becomes:

`t = (x_1)/(5) + (x_2)/(3)`

The distance x₂ depends on x₁:

`x_2 = √((70 - x_1)^2 + 60^2)`

So:
`t = (x_1)/(5) + ( √((70 - x_1)^2 + 60^2) )/(3)`

Find the minimum by setting the derivative to zero:
`(dt)/(dx_1) = 0`

`(dt)/(dx_1) = (1)/(5) - (1)/(3) \frac{70 - x_1} {√(70 - x_1)^2 + 60^2) } = 0`

`70 - x_1 = (3)/(5) √((70 - x_1)^2 + 60^2) \\ \\ (70 - x_1)^2 = (9)/(25)((70 - x_1)^2 + 60^2) \\ \\ (16)/(25)(70 - x_1)^2 = (9)/(25)60^2 \\ \\ (4)/(5)(70 - x_1) = (3)/(5)60 \\ \\ x_1 = 25`

You should stop running 70m - 25m = 45m from b.