a man is in a boat 2 miles from the nearest point on the coast. He is to go to point q, 3 miles down the coast and 1 mile inland. If he can row at 2 mi/h and walk 4 mi/h toward what point on the coast should he row in order to reach point q in the least time?

To find the point on the coast where the man should row in order to reach point q in the least time, we can analyze the time it would take for each combination of rowing and walking distances.

Let's denote the point on the coast where the man should row as point P.

Option 1: Row all the way to point q from the initial position
In this case, the man would row a total of 3 miles down the coast and then walk 1 mile inland. The total time taken would be:
Time = (Distance rowed / Rowing Speed) + (Distance walked / Walking Speed)
= (3 miles / 2 mi/h) + (1 mile / 4 mi/h)
= 1.5 hours + 0.25 hours
= 1.75 hours

Option 2: Row to a point on the coast and then walk to point q
In this case, the man would row to point P on the coast and then walk the remaining distance to point q. Let's denote the distance from the initial position to point P as x miles, which means the distance rowed would be x miles.

To calculate the total time taken, we need to consider the time taken to row to point P and the time taken to walk from point P to point q. The total time taken would be:
Time = (Distance rowed / Rowing Speed) + (Distance walked / Walking Speed)
= (x miles / 2 mi/h) + ((3 - x) miles / 4 mi/h)
= 0.5x hours + 0.75 hours - 0.25x hours
= 0.25x hours + 0.75 hours

To minimize the time, we need to find the value of x that minimizes the total time.

We can plot this as a graph with x on the horizontal axis and time on the vertical axis:

x | Time
---------
0 | 0.75 hours
1 | 1 hour
2 | 1.25 hours
3 | 1.5 hours

From this graph, we can see that the time is increasing as x increases. Therefore, the minimum time would be when x is 0, which means the man should row directly to point q from the initial position.

Therefore, the man should row directly toward point q to reach it in the least time.

To determine the point on the coast from where the man should row in order to reach point Q in the least amount of time, we can apply the principle of optimization by considering the time taken for each possible starting point on the coast.

Let's assume that the man rows from a point A on the coast to point Q. From the given information, we can form a right-angled triangle with point A being the right angle, point Q being the vertex opposite the right angle, and the nearest point on the coast (point B) being the remaining vertex.

We know that the distance from the man's starting point (point A) to point B on the coast is 2 miles, and the distance from point B to point Q along the coast is 3 miles. Additionally, the man needs to travel 1 mile inland from point Q.

Using Pythagoras' theorem, we can find the total distance the man would need to cover if he rows from point A to point Q. The distance will be the hypotenuse of the triangle formed by points A, B, and Q.

The hypotenuse can be calculated as follows:

Distance^2 = (Distance along the coast)^2 + (Distance inland)^2

Distance^2 = (3 miles)^2 + (1 mile)^2

Distance^2 = 9 + 1 = 10

Distance = √10 = approximately 3.162 miles

Now, to find the time taken for the man to row from point A to point Q, we can divide the total distance (3.162 miles) by the rowing speed (2 mi/h):

Time = Distance / Speed

Time = 3.162 miles / 2 mi/h

Time = 1.581 hours (rounded to three decimal places)

To minimize the time taken, the man should row from the point on the coast that minimizes the total distance traveled. In this case, he should row from the point closest to Q along the coast, which is point B.

Therefore, in order to reach point Q in the least amount of time, the man should row from the nearest point on the coast to point Q along the coast directly.

He should row toward the nearest point. Then he can walk the diagonal to point q.