Solve Calc 1 Story Problem: Marathon Runner & Park Trail

[mikky05v]

I'm attempting to help a tutoring student with this problem and I'm having trouble figuring out how to do it.
1. Homework Statement
A marathon runner likes to practice by running in a large park that has a perfectly circular trail, with circumference 50 kilometers. She runs at a constant training speed, and to complete the entire circle takes her 2 hours and 55 minutes. However, it’s her habit that the moment she feels too tired, she stops running on the trail and starts walking on a straight line across the meadow, directly back to her starting point, at a speed exactly 50% of her running speed.

Depending on where she might tire out on the circle, what is the maximum amount of time that she might spend on the run/walk, to the nearest minute?

Let (x,y) =(0,0) be the center of the circular track. Let the runner's starting/ending point be the rightmost point on the circle; that is, (r,0) (in both coordinate systems).

Homework Equations

C=2πr, C= circumference, r= radius
d=s*t, d= distance, s=speed, t=time
a=θ*r, a=arc length, θ=angle, r=radius
d=√[(x₂-x₁)+(y₂-y₁)], d=distance between two points
x=rcosθ, x=rectangular coordinates, (r,θ)=polar coordinates
y=rsinθ, y=rectangular coordinates, (r,θ)=polar coordinates

The Attempt at a Solution

I have no idea how to help him. Can anyone see where to go with this problem? It's been such a long time since I have calc 1 and I am completely lost here.[/B]

 

[RUber]

Start by drawing it out. Your variables are time running and time walking and you want to maximize the sum.

What do you know so far?

 

[mikky05v]

Alright I tried drawing it out but I don't think I see what you're going for here. I found r=25π but I am not seeing what to do with any of the rest of this. What type of problem is this? Maybe I can look up some information on the sections it's into help figure it out. I was thinking optimization maybe but I don't even know if that's a calc 1 subject.

 

[RUber]

Let T be the time before she gets tired.
T is in [0,175] in minutes.
Find a formula for distance from the starting point in terms of T.
Then T2 is the amount of time it takes her to get back. It should be directly related to how far she is.
You should be able to write T2 in terms of T.
Now add them together, and you will have one expression with one variable and you want the max.
Maxes and mins either occur at endpoints or when the first derivative is equal to zero.