Question

Help please ASAP on this question!?

A biatholon is organized for swimmers and speed walkers. The first leg will require swimming from a launch 4 km from the shore of the lake. The second leg involves a walk to the finish line which is 8 km from a point on the shoreline directly across from the launch.

To minimize the length of time of the race, at what point down shore should a particular contestant aim if that person swims at a rate of 3 km/h and walks at a rate of 5 km/h?

Answer 1

3 km away from the point on the shoreline directly across the launch

Explanation:

Let the athlete swim to the point on the shoreline `x` km from the point on the shoreline directly across from the launch, and walk the rest of the way.

The length he swims is `sqrt{x^2+16}` km, and he walks `8-x` km. The total time taken (in hours) is given by

`t = sqrt{x^2+16}/3+{8-x}/5`

To minimize this we need to set the derivative `dt/dx` to zero.

Now

`dt/dx = 1/3 x/sqrt{x^2+16} -1/5`
and setting this to zero leads to

`x/sqrt{x^2+16}=3/5`

which implies that

`x^2/{x^2 +16}=9/25 implies x^2/16 = 9/16 implies x = 3`