Question

brandon is on one side of a river that is 80 m wide and wants to reach a point 200 m downstream on the opposite side as quickly as possible by swimming diagonally across the river and then running the rest of the way. find the minimum amount of time if brandon can swim at 2 m/s and run at 4 m/s.

Answer 1

v1 = 2 mph = .893 m/s

v2 = 4 mph = 1.786 m/s v2 = 2 v1

T = L / v1 + (200 - x) / v2

Where T = total time and L is the distance swum followed by distance x

T = 2 L / 2 v1 + (200 - x) / 2 v1

2 v1 T = 2 L - 200 - x rearranging

2 v1 dT / dt = 2 dL / dt - dx / dt = 0 setting dT / dt = 0

2 dL / dt = dx / dt

dL / dx = 1/2

Now sin θ = x / L = 1/2

So sin 30 = 1/2 and θ = 30 deg

x = 80 tan θ = 80 * .577 = 46.2 m

200 - x = 153 .8 m

t1 = 46.2 / .893 = 51.7 sec

t2 = 153.8 / 1.786 = 86.1 sec

T = 51.7 + 86.1 = 138 sec