Question

A woman at a point A on the shore of a circular lake with radius 2 mi 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 4 mi/h and row a boat at 2 mi/h. How should she proceed

Answer 1

Answer:

She should only walk, and not take the boat ride

Step-by-step explanation:

please see attached a drawing for the problem

we want to minimize the travel time from A to C

given data

Radius of lake= 2mi

walk rate= 4mi/h

boat speed= 2mi/h

The distance from A to B from the drawing shows that

cosθ=adj/hyp=d1/4

d1=4cosθ

The arc length from A to C

S=rθ

but arc length is expressed as 2θ

S=2(2θ)

S=4θ=d2

d2=4θ

we aim to minimize the time of travel

time= distance/speed

t1=time taken from A to B

t1= time taken from B to C

t1=d1/2= 4cosθ/2=2coθ (note speed of boat is 2mi/h)

t2=d2/4=4θ/4=4 (note speed of walk is 4mi/h)

total time =t1+t2

T=2cosθ+θ

the time is a function of θ

T(θ)=2cosθ+θ

since we aim to minimize the travel time, let us find the derivative of the time function and equate it to zero(0)

T'(θ)= -2sinθ+1

0=-2sinθ+1

2sinθ=1

sinθ=1/2

there are two solutions 30° and 150°

but 150° is too much for the angel.

let us try som critical angle between 30° and 90°

like

0= 0 radians

30°= π/6

90°= π/62

for 0 we have

T(0)=2cos0+0

T(0)= 2hours

for 30 we have

T(π/6)=2.25hours

for 90 we have

T(π/2)=1.57hours

since we aim to minimize the travel time, the angle she should follow is 90° from the start point, this clearly shows that she should only walk along the circumference from A to C