A man wants to travel across a river from point a to point D

A man wants to travel across a river from point a to point D. He wants to do this as quickly as possible. His maximum boating speed is 8 km/hr and his maximum running speed is 6 km/hr. He has three options: Drive his boat directly across the river from point A to point B and then run to point D. Drive directly from point A to point to D. Drive to some point C between B and D and then run to D. Which of those options provides him with the minimum time of travel? In particular, where should the point C be chosen to achieve this minimum?

Solution

We need to assume the following:

1. A and B are on a horizontal line, i.e., angle ABC and angle ABD are 90 degrees.

2. The river has no current and hence the effective speed of the boat is the same as its normal speed in still water

SOLUTION

Let the distance from B to C be d km.

By Pythagoras Theorem, AD = sq.rt(10² + 4²) = sq.rt of 116 ............. (1) and

AC = sq.rt(d² + 4²) ....................................(2)

Given, boat speed = 8 kmph ...........................................................(3) and

running speed = 6 kmph ........................................................(4)

Option 1: from A to B by boat and run from B to D

Time required, say t1 = (4/8) + (10/6) = 0.5 + 1.67 = 2.17 hrs or t1² = 4.7089 ....... (5)

Option 2: straight from A to D by boat

Time required, say t2 = distance AD/8 or t2² = 116/64 = 1.8125 ...... .. .... (6) [for 116 see (1)]

Option 3: from A to C by boat and run from C to D

Time required, say t3 = (distance AC/8) + (10 - d)/6 or

t3 = {(d² + 4²)}/8 + (10 - d)/6 ....... (7) [refer (2) for AC and CD distances]

To minimise t3, differentiating (7) w.r.t to d, and equating to 0, we have

2d/{16(d² + 4²)} - (1/6) = 0 or 2d/{16(d² + 4²)} = 1/6 or d/{(d² + 4²)} = 8/6 = 4/3

or squaring both sides, d²/(d² + 4²) = 16/9 or 9d² = 16d² + 256 or d² < 0 which is absurd since a square cannot be negative. Thus, there is no point C by which the time can be minimised.

Hence, the time minimising option is   

Option 2: straight from A to D by boat giving a total time of 1.8125 ~ 1.35 hours ANSWER