Calculus question HELP ASAP!

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

Let P = point on the shore that is 4 km from the starting point offshore

Let Q = point on the shore to which the swimmer swims

 

Let x = distance from P to Q

 

Then 8 - x = distance from Q to the finish line and √(16+x²) = distance from the start to point Q.  

 

Distance = (Rate)(Time), so Time = Distance/Rate

 

T(x) = time it takes if the swimmer swims to point Q

 

T(x) = Time swimming + Time walking

 

T(x) = √(16+x²)/3 + (8-x)/5, where 0 ≤ x ≤ 8

 

T'(x) = (1/3)(1/2)(16+x²)^-1/2(2x) - 1/5

 

        = x/[3√(16+x²)] - 1/5

 

T'(x) = 0 when 5x = 3√(16+x²)

 

                     25x² = 9(16+x²)

 

                     16x² = 144

 

                        x² = 9    So, x = 3

 

T(3) = 8/3 hr ≈ 2.67 hr

 

T(0) = 44/15 hr ≈ 2.93 hr

 

T(8) = √80/3 hr ≈ 2.98 hr

 

To minimize the length of time of the race, a swimmer should swim to a point 3 km from point P and then walk from there 5 km to the finish line.