$\"\"$

The man is in boat $\"\"$ from the nearest point on the coast.

He is to go to a point $\"\"$ , located $\"\"$ down the coast and $\"\"$ inland.

$\"\"$

The man can row at speed of $\"\"$ and walk at $\"\"$ .

Observe the diagram,

$\"\"$ and $\"\"$ .

$\"\"$

$\"\"$ .

Here the time is considered for both the man can row in water and walking time.

The time taken by man to reach the point $\"\"$ is $\"\"$ .

$\"\"$ .

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

The man can reach the point $\"\"$ in minimum time when $\"\"$ .

$\"\"$

$\"\"$ .

$\"\"$

Let $\"\"$ .

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

$\"\"$ .

Since $\"\"$ and $\"\"$ , the solution is in the interval $\"\"$ .

Newton $\"\"$ s approximation method formula : $\"\"$ .

Substitute $\"\"$ and $\"\"$ .

$\"\"$ .

Perform Newton approximation for $\"\"$ .

The calculations for si iterations are shown in the table.

\ \

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Observe the table:

The minimum time approximation is $\"\"$ .

$\"\"$

The minimum time approximation is $\"\"$ .