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
Ratio of the arc length to chord length between two points on the lake is
always lesser than the ratio of the walking and rowing speeds.
She should walk around the half the perimeter of the circular lake from
point A to point C.
Reasons:
Location of woman = Point A
Radius of the lake = 2 miles
Location of the destination = Point B
Distance between point A and point B = Diameter of the lake
Rate of walking = 4 mi/h
Rate at which she rows the boat = 2 mi/h
Required:
How she should proceed
Solution:
If she Journey's by boat;
Distance she would have to row the boat = The diameter of the lake
Distance = Diameter of lake = 2 × Radius
Distance by boat = 2 × 2 miles = 4 miles
[tex]Time = \dfrac{Distance}{Rate \ (speed)}[/tex]
Therefore;
[tex]Time \ by \ boat= \dfrac{4 \ miles}{2 \ miles/hour} = 2 \ hours[/tex]
If she journey's by walking
Distance she would have to walk, D = Half the circumference of the circle
∴ D = π × Radius
Distance she would have to walk, D = π × 2 miles = 2·π miles
[tex]Time \ taken \ to \ walk= \dfrac{2 \cdot \pi \ miles}{4 \ miles/hour} = \dfrac{1}{2} \cdot \pi \ hours \approx 1.571 \ hours[/tex]
Arc length = Radius × θ
Chord length = Radius × 2 × sin(θ/2)
For 0 ≤ θ ≤ π, we have; 0 ≤ sin(θ/2) ≤ 1
From the attached graph, as θ increases, the ratio of chord length to arc
length decreases, therefore, the maximum gain by boat is the direct route,
which takes more time.
Therefore, given that it takes less time walking around the lake, than to
row the boat across the lake, the she should walk around the lake.
Learn more here:
https://brainly.com/question/16597466
