A man in a rowboat at p, see fig 2c, 5km from the nearest point A on the straight shore. He wished to reach a point B, 6km from A along the shore, in a shortest time. Where should he land if he can row 2km/h and walks 4km/h?
P
5 km
A C B
If this is a typical problem, then if he lands at a distance x from A, then the distance on water is
√(x^2+5^2)
and the distance on land is 6-x
So, the time taken is
t = √(x^2+25)/2 + (6-x)/4
dt/dx = x / 2√(x^2+25) - 1/4
= (2x-√(x^2+25)) / 4√(x^2+25)
dt/dx=0 when x=5/√3
at that point, the minimum t = (6+5√3)/4
Unfortunately we cannot see fig 2c unless you describe it or send a link.
i cant attach the triangle...
Don't get it ?? Help!!
To find the point where the man should land to reach point B in the shortest time, we need to consider both his rowing and walking speeds.
First, let's calculate the time it takes for the man to row from point P to any point on the shore. Since the distance from P to A is 5 km and his rowing speed is 2 km/h, the rowing time can be calculated as distance divided by speed: 5 km / 2 km/h = 2.5 hours.
Next, let's calculate the time it takes for the man to walk along the shore from point A to point B. The distance from A to B is given as 6 km and his walking speed is 4 km/h. So the walking time can be calculated as distance divided by speed: 6 km / 4 km/h = 1.5 hours.
To minimize the total time, the man should aim to spend equal time rowing and walking. This means that he should land at a point that is halfway between A and B.
To find the halfway point between A and B, we can calculate the distance from A to the halfway point and add it to the distance from A to P. Since the total distance from A to B is 6 km and the halfway point is at the 3 km mark, the distance from A to the halfway point is 3 km.
Therefore, the man should land at a point 5 km from P (where he is currently located) and 3 km from A. Let's call this point C. It will be the halfway point between P and B, and will ensure that the man spends equal time rowing and walking.
To summarize, the man should land at point C, which is 5 km from P and 3 km from A, in order to reach point B in the shortest time.