Probem46+Alex+G.+Marlee



A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the piont C diametrically opposite A on the other side of the lake in the shortest possible time. She can walk at the rate of 4 mi/h and row a boat at 2 mi/h. How should she proceed?

Split this into the times for walking and for rowing:

since d=rate x time, t=d/r t(boat) = d/2 t(walk) = d/4

Boat: <ABC = 90 degrees and x= segment ab cosø=x/4 ab=4cosø (this is the distance for the boat) t(boat)= (4cosø)/2

Walk: <DBC=90 degrees <BDC=2ø curve BC = 2ø/360 t(walk)=(2(360)r x 2ø/360)/4

soooo

T(total trip)= (2(360)r x 2ø/360)/4) + ((4cosø)/2) take the derivative of this T'=1-2sinø set this equal to 0 This is the angle that the woman should set out at to reach her destination as quickly as possible!
 * ø=(pi)/6**

I would say that the domain of this function, is that 0But what is our answer? What IS the time that it will take for the woman to travel, if she sets of at a 30 degree angle?  Plug in our answer to this equation: T(total trip)= (2(360)r x 2ø/360)/4) + ((4cosø)/2), and get that the total time is 12.06 hours. It turns out that I maximized instead of minimized accidentally. If you plug in 0 for the angle (which means the woman simply rows across the lake, you get that it would take 2 hours. So your choice of time is 2 hours, 1.57 hours, and 12.06 hours. Which x-value yields the minimum time?