Calculus Colorado State University Minimize Travel Time Discussion

Exploring applied optimization problems that minimize travel time.

Part I: Complete the following steps:

1. Read Example 4.34 in Section 4.7 of Calculus, Volume 1.
2. Consider the following scenario:

A lifeguard is at point A of a circular pool with diameter 40 m. He must reach someone who is drowning on the exact opposite side of the pool, at position C. The lifeguard swims with a speed v = 3 m/s from point A to point B, and then runs around the pool from point B to point C at speed w = 9 m/s.

1. Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, ? expressed in radians.
2. Find at what angle ?, in radians, the lifeguard should swim to reach the drowning person in the least amount of time.
3. What is the domain of the function you created in part (a)?

Part II: Based on your work in Part I, discuss the following:

1. How do you know that the function you created in Part I has a maximum and minimum value?
2. Discuss how your answers to Part I would be affected if the diameter of the pool increased.
3. For what running speed would it be faster to swim the entire time? What angle would correspond to this scenario?
4. For what angle, , would it take the longest to reach the drowning person?
5. Suppose the pool was rectangular. Respond to the following:
   1. Does it still make sense to parameterize using ? Why or why not?
      1. If not, what parameter would you use?
      2. If so, how does the parameterization change?
   2. Set up, but do not solve, this problem with a rectangular pool.
6. Answer the following questions that reference Example 4.34:
   1. How do we know that the function T(x) has a maximum and minimum?
   2. What restrictions are there on what the domain of T can be in this scenario?
   3. Elaborate, in your own words, on why we must evaluate T(0) and T(6).