Im stuck and confused?

Question

A woman at a point A on the shore of the circular lake wants to arrive at the point C on the lake diametrically opposite A. She first rows a boat from A to B at a rate of 4 mph and then jogs from B to C at a rate of 8 mph.
The eqn for the time she takes to get from A to C assuming the lake is d miles in diameter is (d/8) (2cos theta + theta) hrs

Questn: If the lake is 4 miles in diameter, what is the least time she can take in getting from A to C via B.
what does it mean by "via B", so why cant it be 1

Answer 1

By "via B" it means that instead of rowing directly to C (where AC is the diameter of the lake) you row to B and jog along the shore to C since AB
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It looks like you have the same problem as the other user. To the other answers, if the angle is in radians then θ is the ratio for an arc legnth which is the distance jogged.

To answer the question, you have a function

T(θ) = ⅛d[2cos(θ) + θ]

To minimize T(θ) take the derivative dT(θ)/dθ, set it to 0 and make sure the second derivative is positive

dT(θ)/dθ = ⅛d - ¼dsin(θ) = 0

sin(θ) = ½

There are 2 solutions for θ

θ = π/6 and θ = 5π/6

Now let's check the second derivative.

dT(θ)/dθ = ⅛d - ¼dsin(θ)
d²T(θ)/dθ² = -¼dcos(θ) = -cos(θ)

So we use our 2 solutions to see if either of them result in a positive second derivative.

d²T(θ)/dθ² = -cos(π/6) < 0
d²T(θ)/dθ² = -cos(5π/6) > 0

So the solution is θ = 5π/6 and the minimum time is

T(5π/6) = ⅛4[2cos(5π/6) + 5π/6]
T(5π/6) = ½[2cos(5π/6) + 5π/6]
T(5π/6) = ½[-√3 + 5π/6]
T(5π/6) = ½[5π/6 - √3]

Answer 2

The equation you cite is suspect. Assuming theta is the angle off the diameter that the woman rows, she would have to run [ theta/360]*pi/8 hours. I dont see a pi in the equation.

The answer can be 1, assuming that theta=0. The question is "can she go from A to C by rowing a shorter distance to B and running the rest at twice the speed to C"?

Answer 3

I don't know the answer to the math question but point a to c via b means that she has to go from point a to point b to point c...she cant go straight from point a to c