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)

Thanks everyone!!

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

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.

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.

Darn.

To determine the point on the coast toward which you should row in order to reach point Q in the least time, we need to find the point P that minimizes the total time taken to row and walk to point Q.

Let's start by visualizing the scenario described in the problem:

1. Draw a straight line representing the coast.
2. Mark a point A on the line, which represents the starting position of the boat, "a" miles from the nearest point on the coast.
3. From point A, draw a straight line perpendicular to the coast, representing the 1 mile distance inland to point Q.
4. Mark a point Q at the intersection of the perpendicular line and the coast, which is "b" miles down the coast and 1 mile inland.

Now, let's apply some mathematical concepts to find the point P toward which you should row to reach point Q in the least time:

1. Recognize that the total time, T, taken to row and walk to point Q can be expressed as the sum of two components: the time taken to row from point A to point P and the time taken to walk from point P to point Q.
T = (distance rowed / speed in rowing) + (distance walked / speed in walking)

2. Since your rowing speed is 2 miles per hour and walking speed is 4 miles per hour, the total time can be simplified to:
T = (distance rowed / 2) + (distance walked / 4)

3. Notice that the distance rowed is the hypotenuse of a right triangle, where the sides are "a" miles and 1 mile. According to the Pythagorean theorem, the distance rowed can be calculated as:
distance rowed = sqrt(a^2 + 1^2)

4. Now that we have expressed the total time in terms of "a", we can differentiate it with respect to "a" and set the derivative equal to zero to find the minimum time. This will give us the value of "a" that corresponds to the point P toward which you should row.

5. Differentiating the equation T = (distance rowed / 2) + (distance walked / 4) with respect to "a":
dT/da = (2a) / [2 * sqrt(a^2 + 1^2)] = a / sqrt(a^2 + 1^2)

6. Setting the derivative equal to zero and solving for "a":
a / sqrt(a^2 + 1^2) = 0
a = 0

7. Note that the result a = 0 does not make sense in this context since it would mean the boat is already at point Q. Instead, we need to consider the critical points by examining the behavior of the derivative.

8. When evaluating the behavior of the derivative, we note that the denominator sqrt(a^2 + 1^2) is always positive, which means that the sign of the derivative will depend on the numerator, "a". Since "a" is greater than zero in this problem, the derivative is positive for all values of "a". This implies that T increases as "a" increases, so there is no minimum.

9. Hence, the point P toward which you should row in order to reach point Q in the least time is when the boat is at the nearest point on the coast.

In summary, you should row directly toward the nearest point on the coast to reach point Q in the least time.