Do dogs know calculus? A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball. Assume C is the point on the edge of the beach closest to the tennis ball (see figure). <IMAGE>




a. Assume the dog runs at speed r and swims at speed s, where r > s and both are measured in meters per second. Also assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball. 

Verified step by step guidance

1. First, understand the problem setup: The dog runs from point A to point D along the beach and then swims from point D to point B in the water. We need to find the total time it takes for the dog to reach the ball at point B.
2. Identify the variables: Let the distance the dog runs along the beach from A to D be 'z - y', where 'z' is the total distance from A to C and 'y' is the distance from C to D. The distance the dog swims from D to B is 'x'.
3. Express the running time: The time taken to run from A to D is given by the formula \( \frac{z - y}{r} \), where 'r' is the running speed.
4. Express the swimming time: The time taken to swim from D to B is given by the formula \( \frac{x}{s} \), where 's' is the swimming speed.
5. Combine the times to form the total time function: The total time \( T(y) \) is the sum of the running and swimming times, which can be expressed as \( T(y) = \frac{z - y}{r} + \frac{x}{s} \).

Key Concepts

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this scenario, the dog’s total time to retrieve the ball is the function to be minimized. By analyzing the relationship between the distances and speeds, we can derive a function that represents the total time, which is essential for determining the optimal path the dog should take.

Functions and Variables

A function is a mathematical relationship where each input is associated with exactly one output. In this problem, T(y) is a function of the variable y, which represents the distance the dog runs along the beach. Understanding how to express the total time as a function of y is crucial for applying calculus techniques to find the minimum time.

Derivatives

Derivatives represent the rate of change of a function and are fundamental in finding local extrema. To minimize the total time function T(y), we will need to compute its derivative and set it to zero to find critical points. This process helps identify the optimal distance y that minimizes the time taken for the dog to reach the ball.