A woman at a point A wants to arrive at the point C ...

Question

A woman at a point A on the shore of a circular lake with a radius r=4 wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time. She can walk at the rate of 8mph and row a boat at 4mph.

What is the longest amount of time it would take her to reach point CC following a path like the path in the figure? 

____ hours

Answer 1

circular lake, radius = 4 miles, diameter = 8 miles

she can row a boat 4 mph, and walk 8 mph

she wants the shortest time to get across the lake which is 8 miles

if she walks around it takes 4pi/4 = pi = about 3.14 hours

if she rows across it takes 8/4 = 2 hours

if she walked half way around that would take 2pi/4 = pi/2 =about 1.57 hours

and then rowed the remaining distance, that would take another 2sqr2/8 = sqr2/4 = .707

= total time of 1.57+.71 = about 2.27

rowing straight across is the shortest time, 2 hours

it might help to provide the diagram

use calculus

total time = distance rowing divided by 4 + distance walking divided by 8

t =r/4 +w/8

find r in terms of w

r = 8 when w = 0

r = 0 when w = 4pi

r=4sqr2 when w= 2pi

find the relation between them

write t in terms of r only

then take the derivative and set = 0

solve for r