A man launches his boat from point A on a bank of a straight river, 1 km wide, a

Question

A man launches his boat from point A on a bank of a straight river, 1 km wide, and wants to reach point B, 1 km downstream on the opposite bank, as quickly as possible (see the figure below). He could row his boat directly across the river to point C and then run to B, or he could row directly to B, or he could row to some point D betweenC and B and then run to B. If he can row 6 km/h and run 8 km/h, where should he land to reach B as soon as possible? (We assume that the speed of the water is negligible compared to the speed at which the man rows.)

Explanation / Answer

Let river's width AC = w
Let the landing point be x km from C
Let the downstream angle (CAX) be a
(speed of running)/ (speed of rowing) = k

distance rowed = sec a

distance run = 1 - tan a

time :
t = sec a - tan a / (4/3)

dt/da = sec a . tan a - 0.75 sec^2 a
      = (sin a - 0.75) / cos^2 a

for minima:

sin a = 0.75

For minimum time :
sin a = w/k
x = w/sqrt(k^2-1)
x = 1/sqrt((8/6)^2 - 1)
= 3/sqrt(7) km
= 1.13 km

That means he should row directly to B.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.