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