A man in rowboat 12km from the nearest point A. On a straight shore wishes to reach point B 16km from A along shore.where should he land if he can row 4km/hr and walks 8km/hr.
He can land anywhere he wants between A and B.
If you want to find the path of least time, however, you need to minimize
√(x^2+12^2)/4 + (16-x)/8
where he lands x km from point A.
To determine where the man should land, we need to find the distance he would have to row and walk to minimize the total travel time. Let's break down the steps to solve this problem:
Step 1: Calculate the time it takes to row to the shore:
Distance to shore = 12 km
Rowing speed = 4 km/hr
Time taken to row to the shore = Distance to shore / Rowing speed
= 12 km / 4 km/hr
= 3 hours
Step 2: Calculate the distance he needs to walk along the shore:
Total distance to be covered = 16 km
Distance already covered by rowing = 12 km
Distance left to cover by walking = Total distance - Distance already covered
= 16 km - 12 km
= 4 km
Step 3: Calculate the time he takes to walk:
Walking speed = 8 km/hr
Time taken to walk = Distance / Walking speed
= 4 km / 8 km/hr
= 0.5 hours
Step 4: Calculate the total time:
Total time taken = Time taken to row + Time taken to walk
= 3 hours + 0.5 hours
= 3.5 hours
So, if the man wants to minimize his total travel time, he should land where he rows to the shore and then walks for 0.5 hours.
Therefore, he should land at a point 12 km from point A along the shore.
To determine where the man should land, we need to find the best point on the shore which minimizes the total time he will take to reach point B.
Let's break the journey into two parts:
1. Rowing from the man's current position to a point on the shore.
2. Walking along the shore to point B.
First, let's calculate the time it would take to row from the man's current position to a point on the shore. The distance from the man to the point on the shore is the straight-line distance, which is 12 km.
Time taken for rowing = Distance / Speed
Time taken for rowing = 12 km / 4 km/hr
Time taken for rowing = 3 hours
Next, let's calculate the time it would take to walk along the shore from the point on the shore to point B. The distance to be covered is the straight-line distance between the two points, which is 16 km.
Time taken for walking = Distance / Speed
Time taken for walking = 16 km / 8 km/hr
Time taken for walking = 2 hours
Therefore, the total time will be the sum of the time taken for rowing and the time taken for walking:
Total time = Time taken for rowing + Time taken for walking
Total time = 3 hours + 2 hours
Total time = 5 hours
To minimize the total time, the man should choose the landing point on the shore such that the time taken for rowing is minimized, because the time taken for walking will be the same regardless of the landing point.
Since the man rows at a constant speed of 4 km/hr, the landing point should be chosen such that the distance to be rowed is minimized. In other words, the landing point should be the closest point on the shore.
Hence, the man should aim to land at the closest point on the shore, which is point A.