You are in a boat "a" miles from the nearest point on the coast. You are to go to a point Q, which is "b" miles down the coast and 1 mile inland. You can row at 2 miles per hour and walk at 4 miles per hour. If a=3, and b=4, toward what point on the coast should you row in order to reach Q in the least time? (Round answer to 3 decimal places)

Question

Thanks everyone!!

Answer 1

if the point on the coast is P, a distance 0<=x<=b down the coast, then the distance traveled (assuming straight cross-country hiking) is

d=√(x^2+3^2) + √((4-x)^2+1^2)
= √(x^2+9) + √(x^2-8x+17)
The travel time is thus

t = √(x^2+9)/2 + √(x^2-8x+17)/4
dt/dx = 4x/√(x^2+9) - 2(4-x)/√(x^2-8x+17)
dt/dx = 0 at x=1.565

Answer 2

The key here is the function of time, one wants to minimize it.

time= time rowing+timewalking
= distancerowing*2 + distancewaking*4
= 2 sqrt(x^2+3^2) + 4*(sqrt(1^2+(4-x)^2)

now examine you diagram, x is the distance down the coast you are aiming for.

dtime/dx=you can do this, set to zero, and find x

It is much more interesting with a current moving along the shore, or a tide going out.

Answer 3

Hmmm. You made the same mistake I did the first time through. time = distance/speed, not distance*speed.

I might have missed it, but the answer I came up with was about 3.58, which seemed awfully close to 4, considering the boat was so much slower than the feet.

Answer 4

Darn.