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
1 answer:
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
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Answer:
Step-by-step explanation:
This is a third degree polynomial since we have 3 zeros. We find these zeros by factoring the given polynomial. The zeros of a polynomial are where the graph of the function goes through the x-axis (where y = 0). If x = -4, the factor that gives us this value is (x + 4) = 0 and solving that for x, we get x = -4. If x = -2, the factor that gives us that value is (x + 2) = 0 and solving that for x, we get x = -2. Same for the 5. The way we find the polynomial that gave us these zeros is to go backwards from the factors and FOIL them out. That means that we need to find the product of
(x + 4)(x + 2)(x - 5). Do the first 2 terms, then multiply in the third.
, which simplifies to
No we multiply in the final factor of (x - 5):
which simplifies to
If you are aware of the method for factoring higher degree polymomials, which is to use the Rational Root Theorem and synthetic division, you will see that this factors to x = -4, -2, 5. If you know how to use your calculator, you will find the same zeros in your solving polynomials function in your apps.