A man is in a boat 4 miles off a straight coast. He wants to reach a point 10 miles down the coast in the least possible time. If he can row 4 miles per hour and run 5 miles per hour, where should he land the boat?
To determine where the man should land the boat to reach a point 10 miles down the coast in the least possible time, we need to consider the combination of rowing and running.
Let's analyze the scenario:
1. Rowing: The man can row at a speed of 4 miles per hour. If he rows directly towards the coast from his current position (4 miles off the coast), it would take him 1 hour to cover the 4-mile distance.
2. Running: The man can run at a speed of 5 miles per hour. Once he reaches the coast, he needs to run along the coast to cover the remaining 10 miles.
Now, let's evaluate the two options:
Option 1: If the man rows to the point 10 miles down the coast (i.e., he rows 10 miles), it would take him 10/4 = 2.5 hours.
Option 2: If the man rows to a point closer to the coast, he would need to run a shorter distance. Let's assume he rows to a point X miles down the coast. The remaining distance he needs to run would be 10 - X miles.
In this case, the time taken to row would be X/4 hours, and the time taken to run would be (10 - X)/5 hours. Therefore, the total time would be (X/4) + (10 - X)/5.
To find the point X where the total time is minimized, we can differentiate the expression (X/4) + (10 - X)/5 with respect to X and equate it to zero.
Differentiating, we get (1/4) - (1/5) = 0.
Solving (1/4) - (1/5) = 0, we find X = 2.5.
This indicates that the man should row 2.5 miles down the coast and then run the remaining 10 - 2.5 = 7.5 miles. This would result in the least possible time.
Therefore, the man should land the boat approximately 2.5 miles down the coast to reach a point 10 miles down the coast in the least possible time.