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**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

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=√[(x

_{2}-x

_{1})+(y

_{2}-y

_{1})], 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]