Question

A person needs to swim to shore from Point A and walk to Point B. They can swim at 2mph and walk at 4 mph. Write an expression for the total time their journey will time their journey will take in terms of x. Then use your graphing calculator to estimate the value of x which will make the trip last a minimum amount of time. Also give the total minimum time for the trip.

Answer 1

I assume the person swims from A to C, which is a distance x from the point D on shore closest to A, then walks from C to B.

Note the lamentable lack of any actual numbers, which limits my ability to provide a numeric answer.

Without loss of generality, let AD=1, and CB = ax for some constant a.

You can scale the diagram as needed

distance from A to C: √(1+x^2)
distance from C to B: ax

time t for the trip

t = 1/2 √(1+x^2) + 1/4 ax
dt/dx = x/[2√(1+x^2)] + a/4
= [2x + a√(1+x^2)] / 4√(1+x^2)

dt/dx = 0 when x = a/(4-a^2)